GREENIEAFS  INTRODUCTORY  ARITHMETIC. 

Improved  Stereotype  Edition. 

INTRODUCTION 

TO    THE 

NATIONAL   ARITHMETIC, 

DESIGNED  FOR  CO:.IMON  SCHOOLS, 


BY  BENJAMIN  GREENIEAF,  A.  M., 

PRINCIPAL  OF  BRADFORD  TEACHERS'  i^' 


BOSTON: 

ROBERT  S.  DAVIS,  A*D  GOULD,  KENDALL,  &  LINCOLN, 

NEW  YORK  COLLINS,  BROTHER,  &  Co, 

PHI 

•  THEK. 

•rally. 

1844. 


EDUCATION 


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GREENLEAF'S  INTRODUCTION, 
Improved  Stereotype  Edition. 

INTRODUCTION 

TO    THE 

NATIONAL  ARITHMETIC, 

ON    THE 

INDUCTIVE   SYSTEM; 

COMBINING   THE 

ANALYTIC  AND  SYNTHETIC  METHODS 

WITH   THE 

CANCELLING    SYSTEM; 

IN   WHICH 

THE   PRINCIPLES    OP   ARITHMETIC   ARE    EXPLAINED   AND 
ILLUSTRATED    IN    A   FAMILIAR   MANNER. 

DESIGNED  FOR  COMMON  SCHOOLS. 


BY  BENJAMIN  vGREENLEAF,  A.  M., 

PRINCIPAL  OP  BRADFORD  TEACHERS'  SEMINARY. 


BOSTON: 

ROBERT  S.  DAVIS,  AND  GOULD,  KENDALL,  &  LINCOLN. 

N.  YORK:  ROBINSON,  PRATT,  &  Co.,  AND  COLLINS,  BROTHER,  k  Co. 

PHILADELPHIA:  THOMAS,   COWPERTHWAIT,   &  Co. 

BALTIMORE:  CUSHINO  &  BROTHER. 

And  sold  by  the  trade  generally. 

1844. 


EDUCATION  LIBS. 


Entered  according  to  Act  of  Congress,  in  the  year  1842,  by 

BENJAMIN   GREENLEAF, 
in  the  Clerk's  office  of  the  District  Court  of  the  District  of  Massachusetts. 

GREENLEAF'S  NATIONAL  ARITHMETIC, 

Forming  a  volume  of  upwards  of  300  pages,  handsomely  printed  on  fine 
paper,  and  bound  in  leather.    Fourteenth,  Improved  Stereotype  Edition. 
Also,  a  COMPLETE  KEY  to  this  work,  designed  for  Teachers  only. 

PUBLISHED  BY  ROBERT  S.  DAVIS,  BOSTON, 
And  sold  by  all  the  principal  Booksellers  throughout  the  United  States. 


This  work,  having  been  adopted  in  many  of  the  best  Schools  in  various  sec 
tions  of  the  country,  is  highly  commended  by  all  intelligent  Teachers  who  have  used  it, 
for  ita  practical  adaptation  to  make  thorough  scholars  in  this  department  of  science. 

RECOMMENDATIONS. 

"Haverhill,  May  22,  1843. 

"B.  GREENLEAF,  Esq.  —  Dear  Sir :  We  have  examined  your  Arithmetics,  the  Na 
tional  and  Introductory,  and  take  pleasure  in  expressing  to  you  our  high  satisfaction 
in  them,  as  superior  to  any  books  in  this  branch  of  education  with  which  we  are  ac 
quainted.  We  are  especially  pleased  with  the  accuracy  and  precision  of  the  defini 
tions,  and  with  the  clearness  and  fullness  of  illustration  by  the  examples.  The  two 
together  seem  to  be  just  what  are  needed,  and  we  are  inclined  to  say  all  that  are 
needed  on  this  subject  in  our  Public  Schools.  In  accordance  with  this  view  of  your 
books,  as  members  of  the  General  School  Committee,  we  have  encouraged  their  use 
in  the  schools  in  this  town. 

(Signed,)  "EDWARD  A.  LAWRENCE,  )  Superintending 

A.  S.  TRAIN,  \  School  Committee." 

"  BENJAMIN  GREENLEAF,  Esq.  —Dear  Sir:  I  regard  your  National  Arithmetic  as 
one  of  the  best  I  have  ever  seen.  Perhaps  the  best  proof  of  the  estimation  in  which 
1  hold  its  merits,  is  the  fact  that  I  use  it  in  the  school  under  my  care. 

"  I  am,  Sir,  very  respectfully  yours,  ROGER  S.  HOWARD, 

Principal  of  the  Latin  High  School. 
"  Newburyport,  May  5,  1843." 

"Having  used  Greenleafs  Arithmetic,  in  the  schools  with  which  I  have  been  con 
nected  for  three  years  past,  1  am  prepared  to  give  it  the  preference  over  any  other 
work  of  the  kind  with  which  I  am  acquainted. 

"  Very  respectfully  yours,  A.  FARWELL, 

Principal  of  Abbott  Female  Academy. 
"  Andover,  June  6,  1843." 

From  H.  Morison,  Esq.,  Professor  of  Mathematics  and  President  of  the 

University  of  Maryland,  Baltimore. 

"  This  is  one  of  the  most  complete  books  of  its  kind,  both  in  the  extent  and  ar 
rangement  of  its  matter,  that  has  yet  appeared.  Combining,  as  it  does,  the  Analytic 
and  Synthetic  methods,  and  abounding  in  familiar  examples,  it  is  admirably  calcu 
lated  to  interest  the  pupil,  and  lead  him,  by  easy  and  progressive  steps,  through  the 
difficulties  of  the  science,  to  its  complete  mastery,  and  full  comprehension.  To  make 
the  work  more  perfect,  than  a  treatise  on  Arithmetic  merely  could  be,  the  author  has 
added  many  geometrical,  mechanical,  philosophical,  and  astronomical  problems,  and 
a  concise  system  of  Book-keeping,  so  that,  without  the  aid  of  any  other  book,  it  is 
calculated  to  make  the  perfect  business  man,  in  all  his  various  departments. 

(Signed,)  "H.  MORISON." 

tr3r-  Other  testimonials  to  the  merits  of  this  work,  will  be  found  in  the  advertis 
ing  sheet,  at  the  end  of  the  volume,  to  which  the  reader  is  referred. 


CAMBRIDGE: 
METCALF  AND  COMPANY 


PRINTERS  TO  THE  UNIVERSITY. 


PREFACE.         /, 


THE  following  treatise  is  intended  for  that  class  of 
pupils,  who  may  not  have  sufficient  time  to  read  the  larger 
work  on  this  science,  published  by  the  author  a  few  years 
since,  and  which  has  had  extensive  circulation. 

It  is  believed,  that  this  book  contains  all,  that  is  neces 
sary  to  prepare  the  young  for  the  common  avocations 
of  life. 

If  the  student  wishes  to  obtain  an  extensive  and  full 
knowledge  of  this  science,  he  can  consult  the  National 
Arithmetic. 

It  has  been  a  great  object  with  the  author  to  render  the 
work  practical ;  how  far  he  has  succeeded,  the  public 
must  judge. 

The  questions  are  original,  although  there  may  be  a 
similarity  between  some  of  these  and  others,  which  are 
before  the  public,  and  which  could  not  be  well  avoided. 

Although  the  author  has  carefully  examined  every 
question,  yet,  it  is  possible,  some  few  mistakes  may  be 
found  in  this  work.  These,  however,  will  be  corrected 
in  a  future  edition. 

With  these  few  prefatory  remarks,  the  author  com 
mends  this  small  volume  to  the  candor  of  an  enlightened 
Public. 

THE   AUTHOR. 

BRADFORD  SEMINARY, 
Nov.  1st,  1842. 


M'770187 


ADVERTISEMENT 

TO    THE 

SECOND  (STEREOTYPE)  EDITION. 


THE  first  edition  of  this  work  having  been  favorably 
received  by  the  public,  the  author  is  now  induced  care 
fully  to  revise  it,  and  make  a  few  additions.  It  is  be 
lieved,  that,  in  the  present  edition,  all  the  answers  to  the 
questions  will  be  found  correct. 

Great  pains  have  been  taken  to  make  the  rules  and 
demonstrations  intelligible. 

In  revising  his  work,  the  Author  has  availed  himself 
of  the  aid  and  suggestions  of  many  practical  teachers  ; 
among  whom  he  would  particularly  acknowledge  his  ob 
ligations  to  two  distinguished  teachers  in  Newburyport, 
David  P.  Page,  Esq.,  of  the  English  High  School,  and 
Mr.  Joseph  Williams,  of  the  Grammar  School. 

BENJAMIN  GREENLEAP. 

BRADFORD  SEMINARY, 
July  1st,  1843. 


CONTENTS. 


PAGE 

SECT.  1.  Numeration 7 

2.  Addition 11 

3.  Subtraction 17 

4.  Multiplication          .......  22 

5.  Division      .....••••27 

6.  Application  of  the  preceding  Rules         ...  35 

7.  Tables  of  Money,  Weights,  and  Measures           .        .  36 

8.  United  States'  Money 42 

9.  Compound  Addition    .......  45 

10.  Compound  Subtraction 50 

11.  Exercises  in  Compound  Addition  and  Subtraction       .  53 

12.  Reduction         . 56 

13.  Reduction  Descending 58 

14.  Reduction  Ascending 59 

15.  Miscellaneous 61 

16.  Compound  Multiplication 63 

17.  Compound  Division    .......  67 

18.  Bills 71 

19.  Fractions 74 

20.  Vulgar  Fractions 76 

21.  Addition  of  Vulgar  Fractions 92 

22.  Subtraction  of  Vulgar  Fractions       ....  94 

23.  Multiplication  of  Vulgar  Fractions      ....  98 

24.  Division  of  Vulgar  Fractions 102 

25.  Exercises  in  Vulgar  Fractions 106 

26.  Decimal  Fractions    .......  108 

27.  Addition  of  Decimals 109 

28.  Subtraction  of  Decimals 110 

29.  Multiplication  of  Decimals Ill 

30.  Division  of  Decimals 113 

31.  Reduction  of  Decimals        .        .        .        .        .        .114 

32.  Exercises  in  Decimals      .•••••  117 

33.  Simple  Interest 118 

34.  Partial  Payments 124 

35.  Commission  and  Brokerage 128 

36.  Insurance  and  Policies « ii9 

37.  Stocks 130 

38.  Banking 130 

39.  Discount 131 

40.  Compound  Interest 132 

41.  Equation  of  Payments        ......  135 

42.  Proportion 137 

43.  Compound  Proportion         ......  143 

44.  Company  Business  .......  145 

45.  Double  Fellowship 146 

46.  Duodecimals 148 

47.  Involution 150 

48.  Evolution,  or  the  Extraction  of  Roots      ...  151 

49.  Extraction  of  the  Cube  Root 156 

•50.  Geometrical  Problems 159 

.  51.  Miscellaneous  Questions     ......  165 


CHARACTERS  USED  IN  THIS  WORK. 

$  Contraction,  for  U.  S.,  United  States'  currency,  and 
is  prefixed  to  dollars  and  cents. 

=  Sign  of  equality  ;  as  12  inches  =  1  foot,  signifies, 
that  12  inches  are  equal  to  one  foot. 

-f-  Sign  of  addition  ;  as  8-j-6=  14,  signifies,  that  8  add 
ed  to  6  is  equal  to  14. 

—  Sign  of  subtraction;  8 — 6=: 2,  that  is,  8  less  6  is 
equal  to  2. 

X  Sign  of  multiplication  ;  as  7X6  =  42,  that  is,  7  multi 
plied  by  6  is  equal  to  42. 

-T-  Sign  of  division  ;  as  42-7-6  =  7,  that  is,  42  divided  by 
6  is  equal  to  7. 

*£•  Numbers  placed  in  this  manner  imply,  that  the  upper 
line  is  to  be  divided  by  the  lower  line. 

:  ::  :  Signs  of  proportion  ;  thus,  2  :  4  ::  6  :  12,  that  is, 
2  has  the  same  ratio  to  4,  that  6  has  to  12  ;  and 

such  numbers  are  called  proportionals. 

15  —  5-}- 3  =13.  Numbers  placed  in  this  manner  show, 
that  5  is  to  be  taken  from  15,  and  3  added  to  the 
remainder.  The  line  at  the  top  is  called  a  vincu- 
lum,  and  connects  all  the  numbers,  over  which  it 

2  is  drawn. 

9     Implies,  that  9  is  to  be  raised  to  the  second  power  ; 

3  that  is,  multiplied  by  itself. 

8     Implies,  that  8  is  to  be  multiplied  into  its  square. 


ARITHMETIC. 


Section  1. 

ARITHMETIC  is  the  art  of  computing  by  numbers.  Its 
five  principal  rules  are  Numeration,  Addition,  Subtrac 
tion,  Multiplication,  and  Division. 


NUMERATION. 

Numeration  teaches  to  express  the  value  of  numbers 
either  by  words  or  characters. 

The  numbers  in  Arithmetic  are  expressed  by  the  fol 
lowing  ten  characters,  or  Arabic  numeral  figures,  which 
the  Moors  introduced  into  Europe  about  nine  hundred 
years  ago  ;  viz.  1  one,  2  two,  3  three,  4  four,  5  five, 
6  six,  7  seven,  8  eight,  9  nine,  0  cipher,  or  nothing. 

The  first  nine  are  called  significant  figures,  as  distin 
guished  from  the  cipher,  which  is,  of  itself,  insignificant. 

Besides  this  value  of  those  figures,  they  have  also  an 
other,  which  depends  on  the  place  in  which  they  stand, 
when  connected  together  ;  as  in  the  following  table. 


NUMERATION.  [SECT.  1. 

| 

1         I 


13 

13 

r* 

13 

2 

0 

GO 

a 

£ 

0 

C3 

a> 

"O 

tn 

o 

13 

!» 

3 

13 

QQ 

CO 

c 

£3 

f-—  < 

B 

c 

o 

C 

/^ 

•"tj 

3 

4J 

•  —  < 

3 

q 

3 

CJ 

a 

SB 

H 

<3 

SB 

H 

p 

SB 

H 

9 

8 

7 

6 

5 

4 

3 

2 

1 

9 

8 

7 

6 

5 

4 

3 

2 

9 

8 

7 

6 

5 

4 

3 

9 

8 

7 

6 

5 

4 

9 

8 

7 

6 

5 

9 

8 

7 

6 

9 

8 

7 

9 

8 

9 

Here  any  figure  in  the  first  place,  reckoning  from 
right  to  left,  denotes  only  its  simple  value  ;  but  that  in 
the  second  place,  denotes  ten  times  its  simple  value  ;  and 
that  in  the  third  place  a  hundred  times  its  simple  value  ; 
and  so  on  ;  the  value  of  any  successive  place  being  al 
ways  ten  times  its  former  value. 

Thus  in  the  number  1834,  the  4  in  the  first  place  de 
notes  only  four  units,  or  simply  4  ;  3  in  the  second  place 
signifies  three  tens,  or  thirty  ;  8  in  the  third  place  signi 
fies  eighty  tens  or  eight  hundred  ;  and  the  1,  in  the  fourth 
place,  one  thousand  ;  so  that  the  whole  number  is  read 
thus,  — one  thousand  eight  hundred  and  thirty-four. 

As  to  the  cipher,  0,  though  it  signify  nothing  of  itself, 
yet,  being  joined  to  the  right  hand  of  other  figures,  it  in 
creases  their  value  in  a  tenfold  proportion  ;  thus  5  signi 
fies  only  five,  but  50  denotes  5  tens  or  fifty  ;  500  is  five 
hundred  ;  and  so  on. 

NOTE.  —  The  idea  of  number  is  the  latest  and  most  difficult  to  form. 
Before  the  mind  can  arrive  at  such  an  abstract  conception,  it  must  be 
familiar  with  that  process  of  classification,  by  which  we  successively 
remount  from  individuals  to  species,  from  species  to  genera,  from 
genera  to  orders.  The  savage  is  lost  in  his  attempts  at  enumeration, 
and  significantly  expresses  his  inability  to  proceed,  by  holding  up  his 
expanded  fingers,  or  pointing  to  the  hair  of  his  head.  See  Lacroix. 


SECT.  1.] 


NUMERATION. 


9 


ENGLISH  NUMERATION  TABLE. 


K  Thousands. 
jg  Tridecillions. 
a  Thousands. 
^  Duodecillions. 
&  Thousands. 
§  Undecillions. 
§5  Thousands. 
g  Decillions. 
§  Thousands. 
§J  Nonillions. 
§  Thousands. 
I  Octillions. 

£§  Thousands. 

1* 

S  Septillions. 

j§  Thousands. 
ij 

5  Sextillions. 

w  Thousands. 

"g 

6  Quintillions. 
j§  Thousands, 
w  Quatrillions. 
J5  Thousands. 

o  Trillions. 

<i 

K  Thousands. 

Xj 

«  Billions. 

3  Thousands. 

"^ 

5  Millions. 

g> 

§  Thousands. 

r  Units. 


To  enumerate  any  number  of  figures, 
they  must  be  separated  by  semicolons 
into  divisions  of  six  figures  each,  and 
each  division  by  a  comma,  as  in  the  an 
nexed  table.  Each  division  will  be 
known  by  a  different  name.  The  first 
three  figures  in  each  division  will  be  so 
many  thousands  of  that  name,  and  the 
next  three  will  be  so  many  of  that  name, 
that  is  over  its  unit's  place.  The  value 
of  the  numbers  in  the  annexed  table  is, 
One  hundred  twenty-three  thousand, 
four  hundred  fifty-six  tridecillions  ;  sev 
en  hundred  eighty-nine  thousand,  one 
hundred  twenty-three  duodecillions  ; 
four  hundred  fifty-six  thousand,  one 
hundred  twenty -three  undecillions;  four 
hundred  fifty-six  thousand,  one  hundred 
twenty-three  decillions  ;  one  hundred 
twenty-three  thousand,  four  hundred 
fifty-six  nonillions  ;  seven  hundred 
eighty-nine  thousand,  seven  hundred 
eighty-nine  octillions  ;  three  hundred 
twenty-three  thousand,  four  hundred 
fifty-six  septillions;  seven  hundred  eigh 
ty-nine  thousand,  seven  hundred  twelve 
sextillions  ;  three  hundred  thirty-three 
thousand,  three  hundred  forty-five  quin- 
tillions  ;  seven  hundred  eighty-nine 
thousand,  one  hundred  twenty-three 
quatrillions  ;  one  hundred  thirty-seven 
thousand,  eight  hundred  ninety  trillions; 
seven  hundred  eleven  thousand,  seven 
hundred  sixteen  billions  ;  three  hundred 
seventy-one  thousand,  seven  hundred 
twelve  millions  ;  four  hundred  fifty-six 
thousand,  seven  hundred  eleven. 

NOTE. — The  student  must  be  familiar  with 
the  names  from  Units  to  Tridecillions,  and  from 
Tridecillions  to  Units,  so  that  he  may  repeat 
them  with  facility  either  way. 


10 


NUMERATION. 


[SECT.  1. 


FRENCH  NUMERATION  TABLE. 


g  Tridecillions. 
a?  Duodecillions. 
§»  Undecillions. 
§  Decillions. 
§  Nonillions. 
J3  Octillions. 
3  Septillions. 
§  Sextillions. 
2  Quintillions. 
JS  Quatrillions. 
§  Trillions. 
S  Billions. 


Millions. 


It  will  be  seen  by  the  annexed  table, 
that  every  three  figures  have  a  different 
name.  Their  value  would  be  thus  ex 
pressed,  Eight  hundred  seventy-six  tri- 
decillions,  seven  hundred  eighty-nine 
duodecillions,  eight  hundred  thirty-five 
undecillions,  one  hundred  twenty-three 
decillions,  three  hundred  sixty-nine  no- 
nillions,  eight  hundred  seventy-three 
octillions,  seven  hundred  seventy-seven 
septillions,  one  hundred  twenty-seven 
sextillions,  eight  hundred  ninety-four 
quintillions,  two  hundred  thirty-seven 
quatrillions,  eight  hundred  sixty-seven 
trillions,  one  hundred  twenty-three  bil 
lions,  six  hundred  seventy-eight  mil 
lions,  four  hundred  seventy-eight  thou 
sands,  six  hundred  thirty-eight. 


So  Thousands. 
§  Units. 
The  pupil  should  write  the  following  numbers  in  words. 

376 

611,711 

3,131,671 

637,313,789 

63,113,716,716 

143,776,711,333 

44,771,631,147,671 

3,761,716,137,716,716 

871  ,  137,637,471  ,378,637 

3,761,716,137,716,167,138 

611,167,637,896,431,617,761,617 

671,386,131,176,378,171,714,563,813 

137,471,716,756,378,817,371,767,386,389,716,473 

NOTE.  —  Although  the  French  method  of  enumeration  is  generally 
used,  yet  it  may  be  well  for  the  pupil  to  understand  both  the  English 
and  the  French. 


SECT.  2.]  ADDITION.  11 

Section  2. 

ADDITION. 

MENTAL  EXERCISES. 

1.  John  had  two  cents  and  Samuel  gave  him  two  more, 
how  many  has  he  ? 

2.  Thomas  had  three    nuts  and  James  gave  him  three 
more,  how  many  has  he  ? 

3.  A  boy  had  four  apples,  and  he  found  two  more,  how 
many  in  all  ? 

4.  I  have  six  dollars,  and  a  man  has  paid  me  three  more, 
how  many  have  I  ? 

5.  Enoch  had  seven  marbles,  and  John  gave  him  two 
more  ;   how  many  has  he  ? 

6.  Benjamin  has  four  dollars,  and  his  sister  has  three  ; 
how  many  have  both  ? 

7.  Paid  five  dollars  for  a  barrel  of  flour,  and  seven  dol 
lars  for  sugar  ;  how  much  for  both  ? 

8.  James  had  two  cents  and  Samuel  gave  him  six  more  ; 
how  many  has  he  ? 

9.  How  many  are  five  apples  and  six  apples  ? 

10.  How  many  are  four  dollars  and  eight  dollars  ? 

11.  How  many  are  2  and  3  ?     2  and  5  ?     2  and  7  ?     2 
and  9  ? 

12.  How  many  are  3  and  3  ?     3  and  5  ?     3  and  7  ?     3 
and  9  ? 

13.  How  many  are  4  and  3  ?     4  and  5  ?     4  and  8  ?     4 
and  9  ? 

14.  How  many  are  5  and  3  ?     5  and  4  ?     5  and  7  ?     5 
and  8  ?     5  and  9  ? 

15.  How  many  are  6  and  2  ?     G  and  4  ?     6  and  3  ?     G 
and  5  ?     6  and  7  ?     6  and  9  ? 

16.  How  many  are  7  and  3  ?     7  and  5  ?     7  and  7  ?     7 
and  6  ?     7  and  8  ?     7  and  9  ? 

17.  How  many  are  8  and  2  ?     8  and  4  ?     8  and  5  ?     8 
and  7  ?     8  and  9  ?     8  and  8  ? 

18.  How  many  are  9  and   1  ?     9  and  3  ?     9  and  5  ?     9 
and  4  ?     9  and  6  ?     9  and  8  ?     9  and  9  ? 

19.  How  many  are  1 1  and  3  ?      1 1  and  2  ?     11   and  4  ? 
11  and  G  ?     11  and  7  ?     1 1  and  9  ?     11  and  11  ?     11  and 


12  ADDITION.  [SECT.  2. 

13?  11  and  12?  11  and  2  and  3  ?  11  and  4  and  4?  11 

and  15  ?  12  and  7  and  3  ?  12  and  6  and  3  ?  8  and  8 

and  4  ?  9  and  5  and  6  ? 

2O.  Gave  nine  cents  for  a  pound  of  cheese,  and  seven 
cents  for  a  quart  of  molasses  ;  what  did  I  give  for  both  ? 
£1.  If  you  buy  a  picture-book  for  eleven  cents,  and  a 

knife  for  nine  cents  ;  what  is  the  cost  of  both  ? 

22.  John  paid  Luke  seven  cents  for  marbles  and  twelve 
cents  for  gingerbread  ;   how  much  money  was  received  ? 

23.  Thomas  paid  twelve  cents  for  a  top  and  eight  cents 
for  cherries  ;  what  did  both  cost  ? 

24.  A  merchant  sold  three  barrels  of  flour  to  one  man 
and  thirteen  to  another  ;  what  was  the  quantity  sold  ? 

25.  I  have  two  appletrees,  one  bears  twelve  bushels  of 
apples,  and  the  other  eleven  ;    how  many  bushels   do 
both  trees  produce  ? 

26.  How  many  are  4  and  2  and  3  ?     5  and  7  and  1  ?     3 
and  4  and  3  ?    6  and  6  and  5  ?    2  and  2  and  8  ?    2  and 

3  and  9  ? 

27.  How  many  are  2  and  6  and  7  ?  2  and  7  and  7  ?  2 
and  8  and  9  ?  2  and  7  and  4  ?  2  and  5  and  9  ?  2  arid 
9  and  6  ?  2  and  3  and  10  ? 

28.  How  many  are  3  and  2  and  2  ?  3  and  3  and  2  ?  3 
and  5  and  5  ?  3  and  4  and  7  ?  3  and  6  and  7  ?  3  and 

7  and  10  ?  3  and  8  and  9  ?  3  and  9  and  9  ? 

29.  How  many  are  4  and  2  and  2  ?  4  and  3  and  3  ?  4 
and  4  and  5  ?  4  and  6  and  7  ?  4  and  7  and  7  ?  4  and 

8  and  3  ?  4  and  9  and  3  ?  4  and  8  and  8  ? 

30.  How  many  are  5  and  3  and  3  ?  5  and  4  and  4  ?  5 
and  5  and  1  ?  5  and  6  and  7  ?  5  and  7  and  8  ?  5  and 

8  and  7  ?  5  and  9  and  9  ?  5  and  10  and  3  ? 

31.  How  many  are  6  and  2  and  7  ?  6  and  3  and  6  ?  6 
and  5  and  4  ?  (J  and  7  and  5  ?  6  and  8  and  7  ?  6  and 

9  and  8  ?  6  and  10  and  10  ? 

32.  How  many  are  7  and  2  and  3  ?  7  and  3  and  3  ?  7 
and  5  and  9  ?  7  and  6  and  6  ?  7  and  8  and  8  ?  7  and 

9  and  8?  7  and  10  and  11  ? 

33.  How  many  are  8  and  2  and  9  ?  8  and  4  and  3  ?  8 
and  7  and  7  ?  8  and  9  and  10  ?  8  and  7  and  9  ?  8  and 

10  and  10  ?  8  and  9  and  12  ? 

34.  How  many  are  9  and  5  and  2  ?  9  and  4  and  3  ?  9 
and  9  and  6  ?  9  and  10  and  3  ?  9  and  8  and  8  ?  9  and 

4  and  9  ?  9  and  9  and  9  f 


SECT.  2.]  ADDITION.  13 

35.  How  many  are  2  and  2  and  4  and  5  ?   3  and  4  and 
5  and  6  ?    4  and  5  and  6  and  7  ?   5  and  5  and  4  and  4  ? 
9  and  1  and  2  and  3  and  5  ? 

36.  James  had  4  apples,  and  Samuel  gives  him  5,  and 
John  gives  him  6  ;  how  many  has  he  ? 

37.  Gave  7  dollars  for  a  barrel  of  flour,  5  dollars  for  a 
hundred  weight   of  sugar,   and  8  dollars  for  a  tub  of 
butter  ;   what  did  I  give  for  the  whole  ? 

38.  Paid  5  dollars   for  a  pair  of  boots,    12  dollars  for 
a  coat,  and  6  dollars  for  a  vest  ;  what  was  the  whole 
cost  ? 

39.  I  have  7  appletrees,  9  cherrytrees,  6  peartrees,  and 
8  plumtrees  ;   how  many  in  all  ? 

40.  In  a  certain  school,  10  scholars  study  grammar,  12 
arithmetic,   7   logic,  2   rhetoric,   and    17   punctuation  ; 
how  many  are  there  in  the  school  ? 

41.  Gave  12  cents  for  an  almanac,  14  cents  for  paper, 
5  cents  for  quills,  and  8  cents  for  an  inkstand  ;  what  did 
I  give  for  the  whole  ? 

42.  Paid  50  dollars  for  a  horse,   and  70  dollars  for  a 
chaise  ;   what  was  the  price  of  both  ? 

43.  A  man  performed  a  journey  in  4  days  ;  the  first  day 
he  travelled  10  miles  ;  the  second   day   12  miles  ;  the 
third  day  12  miles  ;  the  fourth  day  20  miles  ;  what  was 
the  whole  distance  ? 

44.  Paid  2  dollars  for  a  cap,  3  dollars  for  shoes,  7  dol 
lars  for  pantaloons,  6  dollars  for  a  vest,  and  12  dollars 
for  a  coat  ;  what  was  the  cost  of  the  whole  ? 

45.  Gave  75  cents  for  an  arithmetic,  and  25  cents  for  a 
geography  ;   what  was  the  price  of  both  ? 

46.  On  the  fourth  of  July,  20  cents  were  given  to  Emily, 
15  cents  to  Betsey,  10  cents  to  Benjamin,  and  none  to 
Lydia  ;  what  did  they  all  receive  ? 

47.  Bought  four  loads  of  hay  ;   the  first  cost  15  dollars, 
the  second   12   dollars,    the  third   20   dollars,   and   the 
fourth  17  dollars  ;   what  was  the  price  of  the  whole  ? 

The  pupil,  having  performed  the  foregoing  questions, 
will  perceive,  that 

ADDITION  is  the  collecting  of  numbers  together  to  find 
their  sum. 


14  ADD1TIOJN.  [SECT.  2. 


FOR    THE    SLATE 

1.  I  have  three  lots  of  wild  land  ;  the  first  contains  246 
acres,  the  second  764  acres,  and  the  third  918  acres  ; 
how  many  acres  are  there  in  the  three  lots  ? 

OPERATION.  in  tnjg  example,  the  units  are  first 

Acres.  added,   and   their  sum  is  found  to  be 

246  18  ;    in   18  units,   there  are  1  ten  and 

764  8  units;    the  8  is    written   under  the 

918  column  of  units,  and  the  1  (ten)  is  car 

ried  to  be  added  with  the  tens,  which 
1928  Ans.    are   found  to   be  —  1  hundred   and  2 
tens  ;  the  2  is  written  under  the  tens, 
and  the  1  (hundred)   is   carried  to   the   hundreds,  which 
amount  to  19  =   I  thousand  9  hundred  ;    the  whole  of 
which  is  set  down.     Hence  the  propriety  of  the  follow 
ing 

RULE. 

Write  units  under  units,  tens  under  tens,  fyc.  Then  add 
upwards  the  units,  and  if  the  amount  be  less  than  ten,  set 
it  down.  If  the  amount  be  ten  or  more,  write  down  the 
unit  Jigure,  and  carry  the  tens  to  be  added  with  the  columns 
of  tens.  Proceed  in  this  way,  till  the  whole  is  finished, 
writing  down  the  total  amount  in  the  last  column. 

PROOF. 

Begin  at  the  top,  and  add  together  all  the  columns  of 
numbers  downwards,  in  the  same  manner  as  they  were 
before  added  upwards ;  then  if  the  two  sums  agree,  the 
work  is  right. 

QUESTIONS  FOR  THE   SLATE. 


2. 

3. 

4. 

5. 

6. 

7. 

1  1 

47 

127 

678 

789 

1769 

23 

87 

396 

971 

478 

7895 

97 

58 

787 

147 

7  19 

7563 

86 

83 

456 

7  1  6 

937 

8765 

21 7        275        1  766        2512 


ADDITION. 


15 


9. 

789 
567 
743 
435 
678 

3555     3212 

13.          14. 

78956  71678 

37667  12345 

12345  67890 

67890  345  6  7 

78999  89012 

13579  78917 

289436     354409 

17. 

1  7875897 

71675  12 

876567 

98765 

7896 

789 

78 

7 


1922 


11. 

471 

6  1  7 
871 
31  7 
899 


15. 

71123 
45678 
34680 
56777 
67812 
71444 


18. 

789567 
7613 
761 

123123 
70071 
475 
1069 
374176 


16. 

98765 
12345 
67111 
33333 
7  1  345 
99999 


19. 

37 

1378956 

700714 

367 

76117 

46  1  1  779 

9171 

131765 


20. 

895676325678 
123456789012 
876543210988 
7890  12345678 
210987654322 
78901 2345679 
456789012345 
543210987655 
345678901234 
654321098766 
1 04323674322 
210987654321 


21. 


234567891234 
678901234567 
321098765433 
456789012345 
5432  10987655 
789012345678 
2  10987654322 
789012345678 
210987654322 
34567890 1 234 
65432  1  098766 
765432108765 


16  ADDITION.  [SECT.  2. 

22.  What  is  the  sum  of  the  following  numbers,  183,  765, 
838,  375,  857,  and  431  ?  Ans.  3449. 

23.  Add  the  following  numbers,   3791,  83,  71678,  96, 
786,  4711,  and  99.  Ans.  81244. 

24.  Gave  73  dollars  for  a  watch,  15  dollars  for  a  cane, 
119  dollars  for  a  horse,  376  dollars  for  a  carriage,  and 
7689  dollars  for  a  house.     How  much  did  they  all  cost  ? 

Ans.  8272  dollars. 

25.  In  an  orchard,  15  trees  bear  plums,  73  trees  bear 
apples,  29  trees  bear  pears,  and  14  trees  bear  cherries  ; 
how  many  trees  are  there  in  the  orchard  ? 

Ans.  131  trees. 

26.  The  hind  quarters  of  an  ox  weighed  375  pounds  each  ; 
the  fore  quarters  315  pounds  each  ;  the  hide  weighed  96 
pounds,  and  the  tallow  87  pounds.     What  was  the  whole 
weight  of  the  ox  ?  Ans.  1563  pounds. 

27.  A  man  bought  a  farm  for  1728  dollars,  and  sold  it  so 
as  to  gain  375  dollars  ;  how  much  did  he  sell  it  for  ? 

Ans.  2103  dollars. 

28.  A  merchant  bought  five  pieces  of  cloth.     For  the 
first  he  gave  376  dollars  ;   for  the  second   198  dollars  ; 
for  the  third  896  dollars  ;  for  the  fourth  691  dollars  ; 
for  the  fifth  96  dollars.     How  much  did  he  give  for  the 
whole  ?  Ans.  2257  dollars. 

29.  A  merchant  bought  five  hogsheads  of  molasses  for  375 
dollars,  and  sold  it  so  as  to  gain  25  dollars  on  each  hogs 
head  ;  for  how  much  did  he  sell  it  ?      Ans.  500  dollars. 

30.  John  Smith's  farm  is  worth  7896  dollars  ;  he  has  bank 
stock  valued  at  369  dollars  ;  and  he  has  in  cash  850  dol 
lars.     What  is  he  worth  ?  Ans.  9115  dollars. 

31.  Required  the  number  of  inhabitants  in  the  New  Eng 
land  States,   there   being  in  Maine  501,793  ;    in  New 
Hampshire   284,574  ;    in   Massachusetts   737,699  ;    in 
Rhode   Island   108,830  ;    in   Connecticut   309,978  ;    in 
Vermont  291,948.  Ans.  2,234,822. 

32.  Required   the   number  of  inhabitants  in  the  Middle 
States,  there  being  in  New  York  2,428,921  ;  in  New 
Jersey  373,306  ;   in  Pennsylvania  1,724,033  ;   in  Dela 
ware  78,085  ;  in  Maryland  469,232.     Ans.  5,073,577. 

33.  Required  the   number  of  persons   in  the   Southern 
States,  there  being  in  Virginia  1 ,239,797  ;  in  North  Car 
olina  753,419  ;  in  South  Carolina  594,398  ;  in  Georgia 


SECT.  3.]  SUBTRACTION.  17 

691,392  ;  in  Alabama  590,756  ;  in  Mississippi  375,651  ; 
in  Louisiana  35*2,4 11.  Ans.  4,597,824. 

34.  How  many  inhabitants  in  the  Western  States,  there 
being   in  Tennessee  829,210  ;    in  Kentucky   779,828  ; 
in   Ohio    1,519,467  ;    in   Indiana   685,866  ;    in    Illinois 
476,183  ;  in   Missouri  383,702  ;  in  Arkansas  97,574  ; 
in  Michigan  212,267  ?  Ans.  4,984,097. 

35.  How  many  inhabitants  in  the  following  Territories 
and  the  District  of  Columbia,  there   being  in  Florida 
54,477  ;  in  Wisconsin  30,945  ;   in  Iowa  43,112  ;   and  in 
the  District  of  Columbia  43,712  ?  Ans.  172,246. 

36.  How  many  are  the  inhabitants  of  the  United  States, 
there  being  in  New  England  2,234,822  ;   in  the  Middle 
States  5,073,577  ;   in  the  Southern   States  4,597,824  ; 
in  the  Western   States   4,984,097  ;    in   the  Territories 
172,246  ?  Ans.  17,062,566. 


Section  3. 

SUBTRACTION. 

MENTAL  OPERATIONS. 

1.  James  has  three  dollars,  and  John  has  two  dollars  ; 

how  many  has  James  more  than  John  ? 
%•  Thomas  had  five  oranges,  he  gives  two  to  John  ;  how 

many  has  he  left  ? 

3.  Peter  had  six  marbles,  he  gives  two  to  Samuel  ;  how 
many  has  he  left  ? 

4.  Lydia  had  four  cakes,  having  lost  one  ;  how  many 
has  she  left  ? 

5.  Daniel  having  eight  cents,  he  gives  three  to  Mary  ; 
how  many  has  he  left  ? 

6.  Benjamin  had  ten  nuts,   he  gives  four  to  Jane,  and 
three  to  Emily  ;   how  many  has  he  left  ? 

7.  Moses    gives   eleven  oranges  to  John,   and  eight   to 
Enoch  ;  how  many  more  has  John  than  Enoch  ? 

8.  Agreed  to   labor  for  a  man  twelve  days  ?  how  many 
remain,  after  I  have  been  with  him  five  days  ? 


18  SUBTRACTION.  [SECT.  3. 

9.  I  owed  Thomas   nine  dollars,   and   having    paid   him 
seven  ;  how  many  remain  due  ? 

10.  From  ten  dollars,  I  paid  four  dollars  and  three  dol 
lars  ;  how  much  have  I  left  ? 

11.  Timothy  had  eleven  marbles,   he  lost  seven  ;    how 
many  had  he  left  ? 

12.  John  is  thirteen  years  old,  and  his  brother  Thomas 
is  seven  ;  how  much  older  is  John  than  Thomas  ? 

13.  From   15  dollars,   I   paid    five  ;    how  many  have  I 
left? 

14.  Sold  a  barrel  of  flour  for  eight  dollars,  and  a  bushel 
of  wheat  for  two  dollars  ;  what  was  the  difference  in  the 
prices  ? 

15.  Paid  seven  dollars  for  a  pair  of  boots,  and  two  dol 
lars  for  shoes  ;  how  much  did  the  boots  cost  more  than 
the  shoes  ? 

16.  How  many  are  4  less  2  ?    4  less  1  ?    4  less  4  ? 

17.  How  many  are  4  less  3  ?    5  less  1  ?    5  less  5  ? 

18.  How  many  are  5  less  2  ?    5  less  3  ? 

19.  How  many  are  0  less  1  ?     6  less  2  ?     6  less  4  ?    6 
less  5  ? 

20.  How  many  are  7  less  2  ?    7  less  3  ?    7  less  4  ?    7 
less  6  ? 

21.  How  many  are  8  less  6  ?    8  less  5  ?    8  less  2  ?    8 
less  4  ?  8  less  1  ? 

22.  How  many  are  9  less  2  ?    9  less  4  ?    9  less  5  ?    9 
less  7  ?    9  less  3  ? 

23.  How  many  are  10  less  8  ?     10  less  7  ?     10  less  5  ? 

10  less  3?     10  less  1  ? 

24.  How  many  are  11  less  9  ?     11  less  7  ?    11  less  5  ? 

11  less  3  ?     11  less  4  ? 

25.  How  many  are  12  Jess  10  ?     12  less  8  ?    12  less  6  ? 

12  less  4?     12  less  7? 

26.  How  many  are  13  less  11  ?    13  less  10  ?     13  less  7  ? 

13  less  9  ?     13  less  5  ? 

27.  How  many  are  14  less  11  ?     14  less  9  ?     14  less  8  ? 

14  less  6?  14  less  7  ?     14  less  3  ? 

28.  How  many  are  15  less  2  ?     15  less  4  ?    15  less  5  ? 

15  less  7  ?     15  less  9  ?     15  less  13  ? 

29.  How  many  are  16  less  3  ?     16  less  4  ?    16  less  7 

16  less  9?     16  less  11?     16  less  15  ? 


SECT.  3.]  SUBTRACTION.  19 

30.  How  many  are  17  less  1  ?    17  less  3  ?    17  less  5  ? 

17  less  7  ?     17  less  8  ?     17  less  12  ? 

31.  How  many  are  18  less  2  ?    18  less  4  ?    18  less  7  ? 

18  less  8  ?    18  less  10  ?    18  less  12  ? 

32.  How  many  are  19  less  1  ?     19  less  3  ?     19  less  5  ? 

19  less  7  ?    19  less  9  ?    19  less  16  ? 

33.  How  many  are  20  less  5  ?    20  less  8  ?    20  less  9  ? 

20  less  12  ?    20  less  15  ?    20  less  19  ? 

34.  How  many  are  30  less  5  ?    30  less  10  ?    30  less  15  ? 
30  less  20  ?    30  less  25  ? 

35.  Bought  a  horse  for  63  dollars,  and  sold  him  for  70  ; 
what  did  I  gain  ? 

36.  Sold  a  barrel  of  flour  for  8  dollars,  which  cost  me 
10  dollars  ;  what  did  I  lose  ? 

37.  John  travels  25  miles  a  day,  and  Samuel  32  miles  ? 
what  is  the  difference  ? 

38.  I  have  100  dollars,  and  after  I  shall  have  given  17 
to  Benjamin,  and  paid  a  debt  of  30  dollars  to  J.  Smith  ; 
how  many  dollars  have  I  left  ? 

The  pupil,  having  performed  the  above,  will  perceive, 
that 

SUBTRACTION  teaches  to  take   a  less  number   from  a 
greater,  and  to  find  the  difference. 

FOR    THE    SLATE. 

1.  If  I  have  624  dollars  and  lose  342  of  them,  how  many 
remain  ? 


OPERATION.  in  this  question,  we  take  the  2 

From  624  units  from  4  units  and  2  units  remain, 

Take  342  which  we  write  down  under  units, 

-  as  the    first   figure  in  the    answer. 

282  In  attempting  to  take  the  4  tens,  we 

find  a  difficulty,  as  4  cannot  be  taken 

from  2.  We  therefore  borrow  1  (hundred)  from  the  6 
(hundred),  which  being  equal  to  10  tens,  we  add  it  to  the 
2  tens  in  the  upper  line,  making  12  tens,  and  8  (tens)  re 
main,  which  we  set  down.  We  then  proceed  to  the  hun 
dreds.  As  we  have  borrowed  1  from  the  6  hundreds,  the 
6  is  too  large  by  1.  We  must,  therefore,  take  the  3  from 
5,  and  we  find  2  (hundreds)  remain,  which  we  set  down. 


SUBTRACTION. 


[SECT.  3. 


Or  because  the  6  is  too  large  by  1,  we  may  add  1  to  the 
3  and  say  4  from  6  =  2.  This  process  is  called  borrowing 
and  carrying.  Hence  the  following 

RULE. 

Place  the  less  nwnber  under  the  greater;  units  under 
units,  tens  under  tens,  fyc.  Begin  with  the  units ;  and,  if 
the  lower  figure  be  smaller  than  the  upper,  take  it  therefrom, 
and  write  the  difference  below  ;  but,  if  the  upper  figure  be 
less  than  the  lower  figure,  add  ten  to  the  upper  one,  and  place 
the  difference  between  them  under  the  units  as  before,  and 
carry  one  to  the  next  nwnber  at  the  bottom,  and  proceed  thus, 
till  all  the  numbers  are  subtracted. 

NOTE.  The  upper  line  is  called  the  Minuend,  and  the  lower  one 
the  Subtrahend.  The  result  of  the  question  is  called  the  Remainder. 

PROOF. 

Add  the  Remainder  to  the  Subtrahend,  and,  if  their 
sum  be  like  the  Minuend,  the  work  is  right. 


Minuend, 
Subtrahend, 


From 
Take 


QUESTIONS    FOR    THE    SLATE. 

4. 

Miles. 

531 

389 

142 


2. 

£. 
789 
346 

3. 

Cwt. 

376 

187 

443 
6. 

Tons. 
978 

199 

189 

7. 

Gallons. 

67  158 
14339 

779        52819 


1O.  11. 

Miles.  Dollars. 

From         67895      456798 
Take         19999      19089  9 

14. 

Rods. 

From    100200300400500 
Take       908070  6  0  5  0  4  0  3  0 


Minuted. 

76532  1 

1  77777 


5. 

Bushels. 

4789050 

1789582 

2999468 
9. 

Feet. 

1 00000 
90909 


13. 

Socomls. 

555555 

1  77777 


15. 

Acres. 

1000000000000 
9  9  9  999  U 9  9  9  9  9 


SECT.  3.]  SUBTRACTION.  21 

16.  From  1728  dollars,  I  paid  961  dollars  ;    how  many 
remain  ?  Ans.  767  dollars. 

17.  Independence    was   declared   in    1776  ;     how   many 
years  from  this  period  to  the  close  of  the  last  war,  in 
1815  ?  Ans.  39  years. 

18.  The  last  transit  of  Venus  was  1769,   and  the  next 
will  be  1874,  how  many  years  will  intervene  ? 

Ans.  105  years. 

19.  In  1830,  the  number  of  inhabitants  in  Bradford  was 
1856  ;    and   in   1840  it  was  2222  ;    what  was   the  in 
crease  ?  Ans.  366. 

20.  How  many  more  inhabitants  are  there  in  New  York 
city   than   in   Boston,  there  being,   by  the  last  census, 
312,710  inhabitants   in  the    former,  and   93,383  in  the 
latter  city  ?  Ans.  219,327  inhabitants. 

21.  In  1821  there  were  imported  into  the  United  States 
21,273,659  pounds  of  coffee,  and  in  1839,   106,696,992 
pounds  ;  what  was  the  increase  ? 

Ans.  85,423,333  pounds. 

22.  By  the  last  census,  11,853,507  bushels  of  wheat  are 
raised  in  New  York,  and  13,029,756  bushels  in  Pennsyl 
vania  ;  how  many  bushels  in  the  latter  State   more  than 
the  former  ?  Ans.  1,176,249  bushels. 

23.  The  real  estate  of  James  Dow  is  valued  at  3,769 
dollars,  and  his  personal  estate  at  2,648  dollars  ;   he  owes 
John  Smith  1,728  dollars,  and  Job  Tyler  1,161   dollars  ; 
how  much  is  J.  Dow  worth  ?  Ans.  3523  dollars. 

24. 'If  a  man  receive  5  dollars  per  day  for  labor,  and 
it  cost  him  2  dollars  per  day  to  support  his  family  ; 
what  will  he  have  accumulated  at  the  close  of  one 
week  ?  Ans.  18  dollars. 

25.  The  city  of  New  York  owes  9,663,269  dollars,  and 
Boston  owes  1,698,232  dollars  ;  how  much  more  does 
New  York  owe  than  Boston  ? 

Ans.  7,965,037  dollars. 

26.  From  five  hundred  eighty-one  thousand  take  three 
thousand  and  ninety-six.  Ans.  577,904. 

27.  E.  Webster  owns  6,765  acres  of  land,  and  he  gave 
to  his  oldest  brother  2,196  acres,  and  his  uncle  Rollins 
1,981  acres  ;  how  much  has  he  left  ? 

Ans.  2,588  acres. 


MULTIPLICATION. 


[SECT.  4. 


Section  4. 

MULTIPLICATION. 

TABLE  OF  PYTHAGORAS. 


r~-^ 


SECT.  4.]  MULTIPLICATION.  23 


MENTAL  OPERATIONS. 

I.  What  cost  three  bushels  of  wheat  at  three  dollars  per 
bushel  ? 

£•  What  cost  5  barrels  of  flour  at  6  dollars  per  barrel  ? 

3.  What  cost  6  bushels  of  beans  at  2  dollars  per  bushel  ? 

4.  What  cost  5  quarts  of  cherries  at  7  cents  per  quart  ? 

5.  What  will  7  gallons  of  vinegar  cost  at  12  cents  per 
quart  ? 

6.  What  cost  9  acres  of  land  at  10  dollars  per  acre  ? 

7.  If  a  pint  of  currants  cost  4  cents,  what  cost  9  quarts  ? 

8.  If,   in    1   penny,  there  are  4  farthings,  how  many  in 
9  pence  ?    In  7  pence  ?    In  8  pence  ?    In  4  pence  ?    In 

3  pence  ? 

9.  If  12  pence  make  a  shilling,  how  many  pence  in  3 
shillings  ?  In  5  shillings  ?  In  7  shillings  ?  In  9  shillings  ? 

10.  If  4  pecks  make   a  bushel,  how  many  pecks  in  2 
bushels  ?.    In  3  bushels  ?  In  4  bushels  ?  In  6  bushels  ? 
In  7  bushels  ?    In  9  bushels  ? 

II.  If   12   inches  make   1  foot,   how   many  inches  in  3 
feet  ?    In  4  feet  ?    In  5  feet  ?    In  7  feet  ?    In  8  feet  ?    In 
9  feet  ?    In  10  feet  ?    In  12  feet  ? 

12.  If  there  be  9  feet  in  a  square  yard,  how  many  feet 
in  4  yards  ?    In  5  yards  ?    In  6  yards  ?    In  8  yards  ?     In 
9  yards  ?    In  12  yards  ? 

13.  What  cost  3  yards  of  cloth  at  5  dollars  per  yard  ? 

4  yards  ?    5  yards  ?    6  yards  ?    7  yards  ?    8  yards  ?    9 
yards  ?     10  yards  ?     11  yards  ?     12  yards  ?    20  yards  ? 

14.  If  1  pound  of  iron  cost  7  cents,  what  cost  2  pounds  ? 
3  pounds  ?  5  pounds  ?  6  pounds  ?  7  founds  ?  8  pounds  ? 
9  pounds  ?     12  pounds  ? 

15.  If   1    pound    of  raisins   cost   6   cents,  what  cost   4 
pounds  ?    5  pounds  ?    6  pounds  ?    7  pounds  ?    8  pounds  ? 
9  pounds  ?     10  pounds  ?     12  pounds  ? 

16.  In  1  'acre  there  are  4  roods,  how  many  roods  in  2 
acres  I    In  3  acres  ?    In  4  acres  ?    In  5  acres  ?     In  6 
acres  ?    In  9  acres  ? 

17.  A  good  pair  of  boots  is  worth  5  dollars  ;  what,  must 
I  give  for  5  pair  ?  For  6  pair  ?  For  7  pair  ?  For  8  pair  ? 

18.  A  cord  of  good  walnut  wood  may  be  obtained    for 
8  dollars  ;    what  must  I  give  for  4  cords  ?   For  6  cords  ? 
For  9  cords  ? 


..* 


24  MULTIPLICATION.  [SECT.  4. 

19.  A  gallon   of  molasses  is  worth  25  cents,   what   is 
the  value  of  2  gallons  ?    Of  3  gallons  ?    Of  4  gallons  ? 
Of  5  gallons  ?    Of  6  gallons  ? 

20.  What  cost  4  quarts  of  milk  at  5  cents  a  quart  ?    and 
8  gallons  of  vinegar  at  10  cents  a  gallon  ? 

&1.  If  a  man  earn  7  dollars  a  week,  how  much  will  he 
earn  in  3  weeks  ?  In  4  weeks  ?  In  5  weeks  ?  In  6 
weeks  ?  In  7  weeks  ?  In  9  weeks  ? 

22.  If  one  thousand  feet  of  boards  cost  12  dollars,  what 
cost  4  thousand  ?  5  thousand  ?  6  thousand  ?  7  thousand  ? 
12  thousand  ? 

23.  In  1  pound  there  are  20  shillings,  how  many  shil 
lings  in  3  pounds  ?    In  4  pounds  ?    In  6  pounds  ?    In  9 
pounds  ? 

24.  If  3  pair  of  shoes  buy  1  pair  of  boots,  how  many 
pair  of  shoes  will  it  take  to  buy  7  pair  of  boots  ? 

25.  If  5  bushels  of  apples  buy  1  barrel  of  flour,  how 
many  bushels  of  apples  are  equal  in  value  to  12  barrels 
of  flour  ? 

The  foregoing  questions  having  been  performed,  it  will 
be  perceived,  that 

MULTIPLICATION  is  a  compendious  way  of  performing 
Addition,  and  that  it  consists  of  three  parts  ;  the  multi 
plicand,  or  number  to  be  multiplied  ;  the  multiplier,  or 
number  to  multiply  by  ;  and  the  result,  which  is  called 
the  product. 

The  pupil,  having  thoroughly  committed  the  multipli 
cation  Table,  will  notice  the  following 

RULE. 

Place  the  larger  number  uppermost,  and  then  set  the  mul 
tiplier  under  it,  so  that  units  may  be  under  units,  <^*c.,  and 
multiply  by  the  multiplier,  beginning  at  the  unites  place  and 
carry  for  tens  as  in  addition. 

When  the  multiplier  consists  of  more  places  than  one, 
multiply  each  figure  in  the  multiplicand  by  every  figure  in 
the  multiplier,  beginning  with  the  units,  and  placing  Ike 
first  figure  of  each  product  directly  under  its  multiplier, 
then  add  all  their  several  products  together  in  the  same  or 
der,  as  they  stand,  and  their  sum  will  be  the  true  product 
required. 


V 


SECT.  4.] 


MULTIPLICATION. 


25 


When  there  are  ciphers  between  the  significant  figures  of 
the  multipliers,  omit  them,  and  multiply  by  the  significant 
figures  only. 

If  there  be  ciphers  at  the  right  hand  of  the  multiplier  or 
multiplicand,  they  may  be  neglected  in  the  operation,  but 
their  number  must  be  affixed  to  the  product. 

PROOF. 

Multiplication  may  be  proved  by  division,  or  by  multi 
plying  the  multiplier  by  the  multiplicand,  as  in  12th  and 
13th  questions,  or  by  casting  out  the  9's,  thus  ;  cast  the 
9's  from  the  multiplicand  and  place  the  remainder  at  the 
right  hand  of  a  cross,  then  cast  the  9's  from  the  multi 
plier  and  set  the  remainder  at  the  left  hand  of  the  cross, 
then  cast  the  9's  from  the  product,  and  set  the  remainder 
at  the  top  of  the  cross.  Multiply  the  numbers  together 
on  each  side  of  the  cross,  and  cast  the  9's  from  their  pro 
duct,  and  if  the  remainder  be  like  the  number  at  the  top 
of  the  cross,  it  may  be  presumed  the  work  is  right.  See 
question  14. 


QUESTIONS    FOR    THE    SLATE. 


Multiplicand 
Multiplier 


4. 

56807 
5 

284035 

8. 

67895 
36 

407370 
203685 

2444220 


287358 

9. 

78956 
47 

552692 
31  5824 

3710932 

c 


10. 

89325 
91 


11. 

47896 

82 


26 


MULTIPLICATION. 


[SECT.  4. 


12. 

13. 

14. 

7895 
3456 

3456 
7895 

12345 
2231 

47370 
39475 
31580 
23685 

17280 
31104 
2  7  6.4  8 
24192 

12345 
37035 
24690 
24690 

3 

8X6 
3 


27285120   27285120   27541695 


15. 

878532400 
3200 

1  75706480000 
26355972 

281  1303680000 


16. 

713378900 
70080 

57070312000 
49936523 

49993593312000 


Answers. 


17.  Multiply  767853  by  9. 

18.  Multiply  876538765  by  8. 

19.  Multiply  7654328  by  7. 

20.  Multiply  4976387  by  5. 

21.  Multiply  S765448  by  12. 

22.  Multiply  4567839  by  11. 

23.  Multiply  68759  by  5678. 

24.  Multiply  78113  by  70005. 

25.  Multiply  46700  by  60103. 

26.  Multiply  83000  by  10007. 

27.  Multiply  40009  by  40009. 

28.  What  cost  14   barrels   of  apples 
barrel  ? 

29.  What  cost  17  tons  of  hay  at  18  dollars  per  ton  ? 

Ans.  306  dollars. 

30.  What  cost  47  cords  of  wood  at  7  dollars  per  cord  ? 

Ans.  329  dollars. 

31.  What   cost  47  hogsheads  of  molasses  at   13  dollars 
per  hogshead  ?  Ans.  611  dollars. 

32.  What  cost  97  oxen  at  29  dollars  each  ? 

Ans.  2813  dollars. 


6910677. 

7012310120. 

53580296. 

24881935. 

105185376. 

50246229. 

390413002. 

5468300565. 

2806810100. 

830581000. 

1600720081. 

at  3   dollars   per 

Ans.  42  dollars. 


SECT.  5.]  DIVISION.  27 

33.  Sold   a  farm   containing  367  acres,   what  was   the 
amount  at  97  dollars  per  acre  ?       Ans.  35599  dollars. 

34.  An  army  of  17006  men  receive  each  109  dollars  as 
their  annual  pay  ;    what  is  the  amount  paid  the  whole 
army  ?  Ans.  1S53654  dollars. 

35.  If  a  mechanic  deposit  annually  in  the  Savings  Bank, 
407  dollars,  what  will  be  the  sum  deposited  in  27  years  ? 

Ans.  10989  dollars. 

36.  If  a  man  travel  37  miles  in  one  day,  how  far  will 
he  travel  in  365  days  ?  Ans.  13505  miles. 

37.  If  there  be  24  hours  in  one  day,  how  many  hours 
in  365  days  ?  Ans.  8760  hours. 

38.  How  many  gallons  are  in  87  hogsheads,  there  being 
63  gallons  in  each  hogshead  ?  Ans.  5481  gallons. 

39.  If  the  expenses  of  the   Massachusetts   Legislature 
be  1839  dollars  per  day,  what  will  be  the  amount  in  a 
session  of  109  days  ?  Ans.  200451  dollars. 

40.  If  a  hogshead  of  sugar  contains   368   pounds,  how 
many  pounds  in  187  hogsheads  ?     Ans.  68816  pounds. 


Section  5. 

DIVISION. 

MENTAL  EXERCISES. 

1.  A  gentleman  divided  6  apples  between  2  boys  ;  how 
many  did  each  receive  ? 

2.  A  farmer  received  8  dollars  for  2  sheep  ;  what  was 
the  price  of  each  ? 

3.  A  man  gave   15  dollars  for  3  barrels  of  flour  ;  what 
was  the  cost  of  each  barrel  ? 

4.  A  lady  divided  20  oranges  among  her  5  daughters  ; 
how  many  did  each  receive  ? 

5.  If  4  casks  of  lime  cost  12  dollars,  what  is  the  value 
of  1  barrel  ? 

6.  A  laborer  earned  48  shillings  in  6  days  ;  what  did  he 
receive  per  day  ? 

7.  A  man  can   perform  a  certain   piece  of  labor  in  30 
days  ;   how  long  will  it  take  5  men  to  do  the  same  ? 


28  DIVISION.  [SECT.  5. 

8.  When  72  dollars  are  paid  for  8  acres  of  land  ;  what 
cost  1  acre  ?     What  cost  3  acres  ? 

9.  If  21  pounds  of  flour  can  be  obtained  for  3  dollars, 
how  much  can  be  obtained  for  1  dollar  ?    How  much  for 
8  dollars  ?    How  much  for  9  dollars  ? 

10.  Gave  56  cents  for  8  pounds  of  raisins  ;  what  cost 
1  pound  ?    What  7  pounds  ? 

11.  If  a  man  walk  24  miles  in  6  hours,  how  far  will  be 
walk  in  1  hour  ?    How  far  in  10  hours  ? 

12.  Paid  56  dollars  for  7  hundred  weight  of  sugar  ;  what 
cost  1  hundred  weight  ?    What  cost  10  hundred  weight  ? 

13.  If  5  horses  will  eat  a  load  of  hay  in  one  week,  how 
long  would  it  last  one  horse  ? 

14.  In  20,   how  many  times  2  ?     How  many  times  4  ? 
How  many  times  5  ?    How  many  times  10  ? 

15.  In  24  how  many  times  3  ?     How  many  times  4  ? 
How  many  times  6  ?    How  many  times  8  ? 

16.  How  many  times  7  in  21  ?    In  28  ?    In  56  ?    In  35  ? 
In  14  ?    In  63  ?    In  77  ?    In  70  ?    In  84  ? 

17.  How  many  times  6  in  12  ?    In  36  ?    In  18  ?    In  54  ? 
In  60  ?    In  42  ?    In  48  ?    In  72  ?    In  66  ? 

The  pupil  will  now  perceive,  that 

DIVISION  is  a  short  or  compendious  way  of  performing 
Subtraction. 

Its  object  is  to  find  how  often  one  number  is  contained 
in  another.  It  consists  of  four  parts,  the  dividend,  or 
number  to  be  divided  ;  the  divisor,  the  number  we  divide 
by  ;  the  quotient,  which  shows  how  many  times  the  divi 
sor  is  contained  in  the  dividend  ;  and  the  remainder, 
which  is  always  less  than  the  divisor,  and  of  the  same 
name  of  the  dividend. 

I.  When  the  divisor  is  less  than  13,  the  question  should 
be  performed  by 

SHORT  DIVISION. 
1.  Divide  7554  dollars  equally  among  6  men. 

In  performing  this  question,  in- 

quire  how  many  times  6,  the  divi- 
^  ig  contain/d  in  7>  which  is  t 

Quotient      1259         time,  and  1  remaining  ;  write  the 


SECT.  5.] 


DIVISION. 


29 


1  under  the  7,  and  suppose  1,  the  remainder,  to  be  placed 
before  the  next  figure  of  the  dividend,  5  ;  and  the  number 
would  be  15.  Then  inquire  how  many  times  6,  the  divi 
sor,  is  contained  in  15.  It  is  found  to  be  2  times,  and  3 
remaining.  Write  the  2  under  the  5,  and  suppose  the 
remainder,  3,  to  be  placed  before  the  next  figure  of  the 
dividend,  5  ;  and  the  number  would  be  35.  Inquire  again 
how  many  times  35  will  contain  the  divisor,  6.  It  is  found 
to  be  5  .times,  and  5  remaining.  Write  the  5  under  the 
5  in  the  dividend,  and  suppose  the  remainder,  5,  to  be 
placed  before  the  last  figure  of  the  dividend,  4  ;  and  the 
number  would  be  54.  Lastly,  inquire  how  many  times 
54  will  contain  the  divisor,  6.  It  is  found  to  be  9  times, 
which  we  place  under  the  4  in  the  dividend.  Thus  we 
find,  that  each  man  will  receive  1259  dollars. 

From  the  above  illustration  we  deduce  the  following 

RULE. 

See  how  many  times  the  divisor  may  le  contained  in  the 
first  figure  or  figures  of  the  dividend^  and  place  the  result 
immediately  under  that  fgure;  and  what  remains  suppose 
to  be  placed  directly  before  the  next  figure  of  the  dividend ; 
and  then  inquire  how  many  times  these  twofgures  will  con 
tain  the  divisor,  and  place  the  result  as  before ;  and  so  pro 
ceed  until  the  question  is  finished. 


2. 

3)7893762 
2631254 

5. 
6)8765389 

8. 

9)8953784 


3. 

4)4763256 
1  190814 

6. 
7)987635 

9. 

11)7678903 


11.  Divide       479956  by  6. 

12.  Divide       385678  by  7. 

13.  Divide       438789  by  8. 

14.  Divide     1678767  by  9. 

15.  Divide  11497583  by  12. 

c* 


4. 

5)3789565 

7. 
8)378532 

10. 

12)6345321 

Quotients. 

79992| . 

55096f 

54848|, 
186529$. 
95S131J4. 


30  DIVISION.  [SECT.  5. 

16.  Divide  944,580  dollars  equally  among  12  men,  and 
what  will  be  the  share  of  each  ?     Ans.  78,715  dollars. 

17.  Divide  154,503  acres  of  land  equally  among  9  per 
sons.  Ans.  17,167  acres. 

18.  A  plantation  in  Cuba  was  sold  for  7,011,608  dollars, 
and  the  amount  was  divided  among  8  persons.     What 
was  paid  to  each  person  ?  Ans.  876,451  dollars. 

Quotients.  Rem. 

19.  Divide  5678956  by     5.  1. 

20.  Divide  1135791  by     7.  6. 

21.  Divide  1622550  by     8.  6. 

22.  Divide  2028180  by     9.  3. 

23.  Divide  2253530  by  12.  2. 

24.  Divide  1877940  by   11.            9. 

Sum  of  the  quotients,  2084732.         27. 

25.  A  prize,  valued  at  178,656  dollars,  is  to  be  equally 
divided  among  12  men  ;  what  is  the  share  of  each  ? 

Ans.  14,888  dollars. 

26.  Among  7  men,  67,123  bushels  of  wheat  are  to  be 
distributed  ;  how  many  bushels  does  each  man  receive  ? 

Ans.  9,589  bushels. 

27.  If  9  square   feet  make   1   square  yard,   how  many 
yards  in  895,347  square  feet  ?         Ans.  99,483  yards. 

28.  A  township  of  876,136  acres  is  lo  be  divided  among 
8  persons  ;  how  many  acres  will  be  the  portion  of  each  ? 

Ans.  109,517  acres. 

29.  Bought  a  farm  for  5670  dollars,  and  sold  it  for  7896 
dollars,  and  I  divide  the  net  gain  among   6  persons  ; 
what  does  each  receive  ?  Ans.  371  dollars. 

30.  If  6  shillings  make  a  dollar,  how  many  dollars  in 
7890  shillings  ?  Ans.  1315. 

II.  When  the  divisor  exceeds  12,  the  operation  should 
be  performed  by 

LONG  DIVISION, 
as  in  the  following  question. 

31.  A  gentleman   divided    equally   among   his   19  sons, 
4712  dollars  ;  what  is  the  share  of  each  ? 


SECT.  5.]  DIVISION.  31 

OPERATION.  The  object  of  this 

Dividend.  question    is   to    find 

Divisor.  1  9  )  4  7  1  2  (  248  Quotient,  how     many     times 

4712  will  contain  19, 
or  now  many  times 
248  19  must  be  subtract- 

T^Ti  T~i=rTTT  D      -c       e&   from   4712,  un- 
Io24712  Proof.      |U  nothing  ^.^ 

We      first      inquire 

000  Remainder.  how  many  times  19 

may  be  contained  in  47  (thousand).  Having  found  it  to 
be  2  (hundred)  times,  we  write  2  in  the  quotient  and 
multiply  it  by  the  divisor,  19,  and  place  their  product  un 
der  47,  from  which  we  subtract  it,  and  find  the  remainder 
to  be  9,  to  which  we  annex  the  next  figure  in  the  divi 
dend,  1.  And  having  found  that  91  (tens)  will  contain 
the  divisor,  19,  4  (tens)  times,  we  write  4  in  the  quo 
tient,  multiply  it  by  19,  and  place  the  product  76  under 
91,  from  which  we  subtract  it,  and,  to  the  remainder,  15 
(tens),  we  annex  the  last  figure  of  the  dividend,  2,  and 
inquire  how  many  times  152  will  contain  19,  and  we  find 
it  to  be  8  times  ;  and  having  placed  the  product  of  8 
times  19,  that  is,  152,  under  the  152,  we  find  there  is  no 
remainder,  and  that  the  number  4712  will  contain  19,  the 
divisor,  248  times  ;  that  is,  each  man  will  receive  248 
dollars. 

To  prove  our  operation  is  correct,  we  reason  thus.  If 
one  man  receive  243  dollars,  19  men  will  receive  19 
times  as  much,  and  19  times  248  are  4712,  the  same  as 
the  dividend  ;  and  this  operation  is  effected  by  multiply 
ing  the  divisor  by  the  quotient,  and  adding  in  the  remain 
der  if  there  be  one.  The  student  will  now  see  the  pro 
priety  of  the  following 


RULE. 


Place  Ike  divisor  before  the  dividend,  and  inquire  how 
many  times  it,  is  contained  in  a  competent  number  of  figures 
in  the  dividend,  and  place  the  result  in  the  quotient ;  multi 
ply  the  fgure  in  the  quotient  by  the  divisor,  and  place  the 
product  under  those  figures  in  the  dividend,  in  which  it  was 
inquired,  how  many  times  the  divisor  was  contained  ;  sub 
tract  this  product,  from  the  dividend,  and  to  the  remainder 


32  DIVISION.  [SECT.5. 

bring  down  the  next  figure  of  the  dividend  ;  and  then  in 
quire  how  many  times  this  number  will  contain  the  divisor, 
and  place  the  result  in  the  quotient,  and  proceed  as  before, 
until  all  the  figures  of  the  dividend  are  brought  down. 

NOTE  1.  —  It  will  sometimes  happen,  that,  after  a  figure  is  brought 
down,  the  number  will  not  contain  the  divisor  ;  a  cipher  is  then 
placed  in  the  quotient,  and  another  figure  is  brought  down,  and  so 
continue  until  it  will  contain  the  divisor,  placing  a  cipher  each  time 
in  the  quotient. 

NOTE  2.  —  The  remainder  in  all  cases  is  less  than  the  divisor,  and 
of  the  same  denomination  of  the  dividend  ;  and,  if  at  any  time,  we 
subtract  the  product  of  the  figure  in  the  quotient  and  divisor  from  the 
dividend,  and  the  remainder  is  more  than  the  divisor,  the  figure  in 
the  quotient  is  not  large  enough. 

PROOF. 

Division  may  be  proved  by  Multiplication,  Addition,  or 
by  Division  itself. 

To  prove  it  by  Multiplication,  the  divisor  must  be  mul 
tiplied  by  the  quotient,  and,  to  the  product,  the  remainder 
must  be  added,  and,  if  the  result  be  like  the  dividend,  the 
work  is  right. 

To  prove  it  by  Addition.  Add  up  the  several  products 
of  the  divisor  and  quotient  with  the  remainder,  and,  if  the 
result  be  like  the  dividend,  the  work  is  right. 

To  prove  it  by  Division  itself.  Subtract  the  remainder 
from  the  dividend,  and  divide  this  number  by  the  quotient, 
and  the  quotient  found  by  this  division  will  be  equal  to 
the  former  divisor,  when  the  work  is  right. 


83) 

32. 

1486 

83* 

7 
* 

8( 

# 

1 

4 

1791 

83 

33. 

427)567896(1 

427 

329 

656 

58  1 

5373 

328 

1 

1 

4 
g 

08 

81 

757 
747 

I 

4 

8653 
25 

Proof. 

J 

27 
85 

9 
4 

1 

08 
83* 

1 

48678 

4256 
3843 

25*  413 

148678  *  NOTE.     The  aslerisms  show  the  numbers  to 

be  added. 


SECT.  5.] 


DIVISION. 


34. 

144)13824(96 
1296 

~~8~64 
864 


35. 

96)13824(144 
96 


NOTE.    The  34th  question  is  proved  by 
the  35th. 


36. 

86  1  000  )  8963  |  486  (104 
86 


37. 

Rem. 

1  ]  0000  )  7  |  8967  (  7  Quotient. 


38.  Divide 

39.  Divide 

40.  Divide 

41.  Divide 

42.  Divide 

43.  Divide 
Divide 
Divide 
Divide 
Divide 


44. 
45. 
46. 
47. 


48.  Divide 

49.  Divide 


867532 

167008 

345678 

6789563 

78112345 

34533669 

99999999 

47856712 

13112297 

10000000 

15678953 

71800100 


by  59. 
by  87. 
by  379. 
by  1234. 
by  8007. 
by  9999. 
by  3333. 
by  1789. 
by  8900. 
by  7007. 
by  8790. 
by  4701. 


Quotients. 

14703. 

1919. 

912. 


Remainders. 

55. 

55. 

30 

95. 

4060. 

7122. 

0. 

962. 
2597. 
1011. 
6383. 
1727. 


III.  To  multiply  by  a  fraction. 


RULE. 

Multiply  the  given  number  by  the  numerator  of  the  frac 
tion,  and  divide  the  product  by  the  denominator.  If  any 
thing  remain  place  it  over  the  divisor  at  the  right  hand 
of  the  quotient. 

NOTE.  When  the  number  is  such,  that  it  may  be  divided  by  the 
denominator  without  a  remainder,  the  better  way  is  to  divide  the  given 
number  by  the  denominator,  and  multiply  the  quotient  by  the  numer 
ator.  This  is  the  analytical  method. 


34  DIVISION.  [SECT.  5. 

50.  What  is  f  of  144  ? 

Synthetic  method.  Analytical  method. 

144  4)144  Divide  by  4  to 

3  3  Q  get     one     fourth, 

4 )  432  3  and  multiply  by  3 

HTSAns.  T08Ans.  to  get  3  fourths. 

51.  What  is  %  of  365  ?     Ans.  228$. 

52.  What  is  f  of  128  ?     Ans.  54f . 

53.  What  is  ft  of  386  ?     Ans.  210T6T. 

54.  Sold  a  farm  for  1728  dollars  ;   and,  if  I  give  ^  of 
this  sum  to  indigent  persons,  what  do  they  receive  ? 

Ans.  720  dollars. 

55.  If  from  1000  dollars  f  be  taken,  what  sum  will  re 
main  ?  Ans.  625  dollars. 

IV.  To  divide  by  a  fraction. 

RULE. 

Multiply  the  given  number  by  the  denominator,  and  di 
vide  the  product  by  the  numerator. 

56.  Divide  125  by  f 

125  In  this  example,  we  mul- 

8  tiply  by  8  to  reduce  the  125 

5)1  000  to    eighths  ;    and   then   we 

— o  n  n  see  how  often  5  (eighths) 

are  contained  in  them. 

57.  Sold  f-  of  a  house  for  3227  dollars  ;  what  was  the 
value  of  the  whole  house  ?  Ans.  3688  dollars. 

V.  To  divide  by  a  composite   number,  that  is,  a  num 
ber,  which    is   produced    by  the  multiplying   of  two  or 
more  numbers. 

RULE. 

Divide  the  dividend  ~by  one  of  these  numbers,  and  the  quo 
tient  thence  arising  by  another,  and  so  continue ;  and  the 
last  quotient  ivill  be  the  answer. 

NOTE.  To  find  the  true  remainder,  we  multiply  the  last  remainder 
by  the  last  divisor  but  one,  and  to  the  product  add  the  next  preced- 


SECT.  6.]  DIVISION.        |  35 

ing  remainder;  we  multiply  this  product  by  the  next  preceding  divis 
or,  and  to  the  product  add  the  next  preceding  remainder ;  and  so  on 
until  we  have  gone  through  all  the  divisors  and  remainders  to  the  first. 

58.  Divide  67872  by  24. 

OPERATION.  jn  this  question,  we  divide 

4)67872  by  4  and  6,  because  they  are 

6  )  1  69  68  tn®  factors>  or  composite  num- 

-2828  bers  of  24' 

Quotients. 

59.  Divide  765325  by  25  =  5x5.        30613. 
GO.  Divide  123396  by  84  =  7  X  12.        1469. 

61.  Divide  611226  by  81=9x9.         7546. 

62.  Divide  987625  by  125  =  5  X  5  X  5.      7901. 


Section  6. 

APPLICATION  OF  THE  PRECEDING  RULES. 

1.  A  farmer  bought  5  yoke  of  oxen  at  87  dollars  a  yoke  ; 
37  cows  at  37  dollars  each  ;    89  sheep  at  3  dollars  a 
piece.     He  sold  the  oxen  at  98  dollars  a  yoke  ;   for  the 
cows  he  received  40  dollars  each  ;    and,  for  the  sheep, 
he  had  4  dollars  a   piece  ;    what   did   he    gain   by  his 
trade  ?  Ans.  255  dollars. 

2.  In  4008  hours,  how  many  days  ?          Ans.  167  days. 

3.  In  169  weeks,  how  many  days  ?         Ans.  1183  days. 

4.  If  12  inches  make  a  foot,  how  many  feet  in  48096 
inches  ?  Ans.  4008  feet. 

5.  In  15300  minutes,  how  many  hours  ? 

Ans.  255  hours. 

6.  If  144  inches  make   1   square  foot,  how  many  square 
feet  in  20736  inches  ?  Ans.  144  feet. 

7.  An  acre  contains  160  square  rods  ;  how  many  in  a 
farm  containing  769  acres  ?  Ans.  123040  rods. 

8.  A  gentleman  bought  a  house  for  three  thousand  forty- 
seven  dollars,  and  a  carriage  and  span  of  horses  for  five 
hundred  seven  dollars.     He  paid  at  one  time,  two  thou 
sand  seventeen  dollars,  and   at  another  time,  nine  hun 
dred  seven  dollars.     How  much  remains  due  ? 

Ans.  630  dollars. 


36 


NEY  ANl)  WEIGHTS. 


[SECT.  7. 


9.  The  erection  of  a  factory  cost  68,255  dollars  ;  suppos 
ing  this  sum  to  be  divided  into  365  shares,  what  is  the 
expense  of  each  ?  Ans.  187  dollars. 

10.  A  gentleman,  possessing  an  estate  of  375,846  dol 
lars,   bequeathed  7,494   dollars  to  the  Bible    Society  ; 
4,230  dollars  for  the  support  of  schools  ;  and  one  third 
to  his  wife  ;    the  remainder  was  to  be  equally  divided 
among  his  12  sons  and  8  daughters  ;  what  sum  will  each 
receive  ?  Ans.  11,942  dollars. 

11.  There  were  distilled  in  the  United  States  in  1840, 
thirty-six   millions  three   hundred   forty-three   thousand 
two  hundred  thirty-six  gallons  of  ardent  spirits  ;  and  the 
number  of  free  white  males,  over  15  years,  is  four  mil 
lions  seventy-four  thousand  nine  hundred  fifteen  ;   now 
supposing  the  liquor  to  be  drank  by  one  third  of  those 
persons  in  one  year,  what  quantity  would  each  consume? 

Ans.  More  than  26  gallons. 

12.  A  man  gave  half  of  his  estate  to  his  wife,  one  third 
of  what  remained  to  his  son,  and  the  residue  was  equal 
ly  divided  among  his  7  daughters,  who   received  each 
124  dollars  ;  what  was  the  whole  estate  ? 

Ans.  2,604  dollars. 


Section  7. 

TABLES    OF    MONEY,    WEIGHTS, 
AND    MEASURES. 


10  Mills 
10  Cents 
10  Dimes 
10  Dollars 

Mills. 

10  = 

100  = 

1000  = 

10000  = 


UNITED  STATES'   MONEY. 

make         1  Cent,  marked 

1  Dime, 

"  1  Dollar,  " 

1  Eagle, 


Centa. 
1 

10 

100 
1000 


Dimes. 

1 

10 

100 


Dollars: 
=  1 

=     10 


C. 

d. 

I: 


Eaelea. 
1 


SECT.  7.] 


MONEY   AND    WEIGHTS. 


37 


4  Farthings 
12  Pence 
20  Shillings 

qrs. 

4        = 

48        = 
960        = 


ENGLISH  MONEY. 

make  1  Penny, 

"  1  Shilling, 

"  1  Pound, 

d. 
1 

12  =          1 

240  =        20 


marked*  d. 

"        ks. 
£. 


FRENCH  MONEY. 
100  Centimes  make  1  Franc  =  .1875  dollar. 


TROY  WEIGHT. 


24  Grains  make 

20  Pennyweights     " 
12  Ounces  " 


Pennyweight,   marked  dvvt. 
Ounce,  "        oz. 

Pound,  "       Ib. 


grs. 

24 

480 
5760 


dwt. 

1 

20 
240 


oz. 

1 

12 


By  this  weight  are  weighed  gold,  silver,  and  jewels. 

NOTE.  "  The  original  of  all  weights,  used  in  England,  was  a  grain 
or  corn  of  wheat,  gathered  out  of  the  middle  of  the  ear  ;  and  being 
well  dried,  32  of  them  were  to  make  one  pennyweight,  20  penny 
weights  one  ounce,  and  12  ounces  one  pound.  But,  in  later  times,  it 
was  thoutrht  sufficient  to  divide  the  same  pennyweight  into  24  equal 
parts,  still  called  grains,  being  the  least  weight  now  in  common  use; 
and  from  hence  the  rest  are  computed." 


APOTHECARIES'  WEIGHT. 


20  Grains 
3  Scruples 
8  Drams 

12  Ounces 

&     = 

60        = 

480        = 
5760        = 


make 


1 
3 

24 

288 


1  Scruple, 
1  Dram, 
1  Ounce, 
1  Pound, 

dr. 

•J 

=        8 
=      96 


marked 


sc.  or  9. 
dr.  or  3. 
oz.  or  § . 
Ib.  or  Ib. 


oz. 

1 

12      = 


Apothecaries  mix  their  medicines  by  this  weight  ;  but 
buy  and  sell  by  Avoirdupois.  The  pound  and  ounce  of 
this  weight  are  the  same  as  in  Troy  Weight. 


WEIGHTS    AND    MEASURES. 


[SECT.  7. 


AVOIRDUPOIS  WEIGHT. 

make  1  Ounce, 
"  1  Pound, 
"  1  Quarter,  " 

"      1  Hundred  weight/' 


fl6  Drams 
16  Ounces 
28  Pounds 

4  Quarters 
20  Hundred  weight    "      1  Ton, 


marked  oz. 
"       Ib. 
qr. 
cwt. 
ton. 


dr. 

16 

256 

7168 

'  28672 
573440 


oz. 

1 

16 

448 

1792 

35840 


1 

28 

112 

2240 


4      =      1 

80    =20 


=1 


By  this  weight  are  weighed  almost  every  kind  of  goods, 
and  all  metals  except  gold  and  silver.  By  a  late  law  of 
Massachusetts,  the  cwt.  contains  100  Ibs.  instead  of  112 
Ibs. 

LONG  MEASURE. 


12  Inches                    make  1  Foot,             marked     ft. 

3  Feet 

"     I  Yard, 

yd 

5J  Yards,  or  16J  feet 

"     1  Rod,  or  pole 

rd. 

40  Rods 

"     1  Furlong, 

fur. 

8  Furlongs 

"     1  Mile, 

m. 

3  Miles 

"     1  League, 

lea. 

69J  Miles  (nearly) 
360  Degrees 

"     I  Degree, 
"     1  Circle  of  the 

"Deg.  or°. 
Earth. 

in.                         ft. 

12    =          1 

yd. 

36    =          3      = 

1                      rd. 

198    =         16^    = 

5i    =        1 

for. 

7920    =      660      = 

220      =      40    : 

=      1 

63360    =    5280     = 

1760      =    320 

=    8    =    1 

2J  Inches 
4    Nails 

4  Quarters 
3    Quarters 

5  Quarters 


CLOTH  MEASURE. 


make 


Nail,  marked  na. 

Quarter  of  a  yard  "       qr. 
Yard,  "      yd. 

Ell  Flemish,         "     E.  F. 
Ell  English,          "     E.E. 


NOTE.    The  Ell  French  is  not  in  use. 


SECT.  7.] 


WEIGHTS    AJND    MEASURES. 


39 


SQUARE  MEASURE. 


144    Square 
9    Square 
301  Square 
272J  Square 
40    Square 
4   Roods 
640  Acres 


inches        make  1  Square  foot,     marked  ft. 

feet 

yards 

feet 

rods  or  poles  " 


Square  yard, 
1   Square  rod  or  pole," 
1  Square  rod  or  pole, ' c 
1  Rood, 


1  Acre, 

1  Square  mile, 


yd. 
P- 

••  fi. 
"  A. 
"S.M. 


in. 

144  = 
1596  = 
39204  = 
1568160  = 
6272640  = 
4014489600  =  27878400 =3097600  =  102400  =  2560  =  640  =  1 


272J  = 
10890  = 
43560  = 


yd. 
30  J  = 

1210  = 

4840  = 


Pi 

40  = 
160  = 


R. 
1 
4  :=: 


A. 
1  S.M. 


DRY  MEASURE. 


2  Pints 
4  Quarts 
2  Gallons 
4  Pecks 
36  Bushels 


Pt8 
16 
64 
2304       = 


make 

1  Quart,            marked 

qt. 

<( 

1  Gallon, 

cc 

gal 

cc 

1  Peck, 

(C 

pk. 

(( 

1  Bushel, 

cc 

bu. 

(C 

1  Chaldron, 

cc 

ch. 

gal. 

1 

pk. 

2 

= 

i           bUi 

==          8 


288      =      144      =    36        = 


ch. 


This  measure  is  applied  to  all  Dry  Goods,  as  Corn, 
Fruit,  Salt,  Coals,  &c.  A  Winchester  Bushel  is  18J 
inches  in  diameter,  and  8  inches  deep.  The  standard 
Gallon  Dry  Measure  contains  268$  inches. 


ALE  AND  BEER  MEASURE. 


2  Pints 

4  Quarts 
36  Gallons 
54  Gallons 

2  Hogsheads 

2  Butts 


make 

(C 
(C 


Quart,  marked  qt. 

Gallon,  "         gal. 

Barrel,  "         bar. 

Hogshead,  hhd. 

Butt,  "         butt. 

1  Tun,  "         tun. 


40 


MEASURES    AND   TIME. 


[SECT.  7. 


P2 
8 

288 
432 
864 


qt. 

4 

144 
216 
432 


f 

36 

54 

108 


bar. 

1 


hhd. 
=      1 

=    2 


butt. 
1 


NOTE.  By  a  law  of  Massachusetts,  the  barrel  for  Cider  and  Beer 
shall  contain  32  Gallons.  The  Ale  Gallon  contains  282  cubic  or  solid 
inches. 

WINE  MEASURE. 


4  Gills 

make 

2  Pints 

n 

4  Quarts 

ft 

42  Gallons 

(t 

63  Gallons 

tt 

2  Tierces 

tt 

2  Hogsheads 
2  Pipes  or  4  Hhds. 

tt 
tt 

pis.                   qt. 

2  =        1 

gal. 

8  =        4  = 

1 

336  =     168  == 

42  = 

504  =    252  = 

63  = 

672  =     336  = 

84  = 

1008  =     504  = 

126  = 

2016  =  1008  = 

252  = 

1   Pint, 

marked  pt. 

1   Quart, 

qt. 

1  Gallon, 

gal. 

1  Tierce, 

tier. 

1  Hogshead, 

hhd. 

1  Puncheon, 

pun. 

1  Pipe  or  Butt, 

"          pi. 

1  Tun, 

"         tun. 

tier. 

I          hhd. 
:    1  2  : —    ^  pun. 

84  =  2   =  1J=  1         pl. 
126  =  3   =  2   =  11=  1        tun. 
252  =  6=4=3=2-=! 

NOTE.  The  Wine  Gallon  contains  231  cubic  inches.  We  have  no 
statute  specifying  how  many  gallons  a  hogshead,  tierce,  or  pipe,  shall 
contain. 

OF  TIME, 
make 


60  Seconds,  or  60" 

60  Minutes  " 

24  Hours 

7  Days 

4  Weeks 
13  Months,  1  day,  6  hours,  or 

365  days,  6  hours 
12  Calendar  months         " 


1  Minute, 

marked  m. 

1  Hour, 

"       h. 

1  Day, 

"       d. 

1  Week, 

"       w. 

1  Month, 

"       mo. 

I  Julian  Year 

,    "     y- 

1  Year, 

"    y- 

SECT.  7.] 


TIME    AND    MEASURE. 


41 


sec. 

60 
3600 
86400 
604800 
2419200 
31557600 

= 

m. 
1 

60 
1440 
10080 
40320 
525960 

= 

h. 

1 

24 

168 
672 
8766 

= 

d. 

1 

7 
28 
365J 

< 

w. 

=  1 

—  .  4 

=  yi 

NOTE.  The  true  solar  year  is  the  time  measured  from  the  sun's 
leaving  either  equinox  or  solstice,  to  its  return  to  the  same  again.  A 
periodical  year  is  the  time  the  earth  revolves  round  the  sun,  and  is 
365d.  6h.  9m.  14£sec.  and  is  often  called  the  Sidereal  year.  The 
civil  year  is  that,  which  is  in  common  use  among  the  different  na 
tions  of  the  world,  and  contains  365  days  for  three  years  i«  succes 
sion,  but  every  fourth  year  it  contains  366  days.  When  any  year 
can  be  divided  by  four,  without  any  remainder,  it  is  leap  year,  and 
has  366  days.  The  days  in  each  month  are  stated  in  the  following 
distichs. 

Thirty  days  hath  September, 

April,  June,  and  November; 

All  the  rest  have  thirty-one, 

Except  February  alone, 


Which  hath  but  twenty-eight, 
Except  leap  year,  when  it  hath  twenty-nine. 

Or, 

w.      d. 

52     1 

h. 

6    = 

m. 

13 

d.      h. 

1     6     = 

1 

Julian  Year. 

But, 

day. 

365 

h. 

5 

m. 

48 

sec. 

57     = 

1 

Solar  Year. 

And, 

day. 

365 

h. 

6 

m. 

9 

sec. 

1 

Sidereal  Year. 

CIRCULAR  MOTION. 

60  Seconds  make     1  Prime  minute3     marked  '. 

60  Minutes  "         1  Degree,  "       °. 

30  Degrees  "         1   Sign,  "       s. 

12  Signs,  or  360  Degrees,  the  whole  great  Circle  of  the 
Zodiac. 


25 

100 

10 


Inches 
Links 
Links 
Chains 


MEASURING  DISTANCES, 
make 


8        Furlongs 


Link. 

Pole. 

Chain. 

Furlong. 

Mile. 


42 


UNITED   STATES7  MONEY. 


[SECT.  8. 


SOLID  MEASURE. 

1728  Inches  make 

27  Feet 

40  Feet  of  Timber 
128  Feet,  i.  e.  8  feet  in  length,  )  (( 
4  in  height,  and  4  in  breadth,  j 


Foot. 
Yard. 
Ton. 


I  Cord  of  Wood. 


Section  8. 
UNITED    STATES'    MONEY. 

ADDITION. 

RULE.  Place  dollars  under  dollars,  dimes  under  dimes, 
cents  under  cents,  and  mills  under  mills,  and  add  the  col 
umns  together,  as  in  the  addition  of  simple  numbers,  and 
place  a  period  or  point  immediately  after  the  dollars,  sep 
arating  them  from  the  cents. 

NOTE.  The  eagles  and  dollars  are  usually  written  together ;  as 
are  also  the  dimes,  cents,  and  mills.  The  dollars  are  separated  from 
the  cents  by  a  point;  all  the  figures  at  the  left  of  the  point  are  dollars, 
and,  at  the  right  of  the  point,  the  first  two  figures  are  cents,  and  the 
third  is  mills.°Three  dollars  fifteen  cents  six  mills  are  written  $3.150. 
Seventy-four  dollars  three  cents  four  mills  are  written  $  74.034. 
Seventeen  dollars  five  mills  are  written  $  17.005. 


1. 

E.  8.  d.  els.  m. 

7.  5.  6.  4.  3 
1.  6.  8.  9.  7 
4.3.8.  1.6 
5.  8.  3.  1.  3 


$.   eta.  m. 

7  5.  6  4  3 
16.897 
4  3.  8  1  6 
58.313 


3. 

$.   cts.  m. 

1  6.  7  0  5 
14.003 
1  8.  7  1  9 
97.009 


4. 

8.     Ct3. 

1  4  7.  8  6 

789.58 
496.37 
91  1.34 


1  9.  4.  6.  6.  9   194.669   1  4  6.  4  3  6   2  3  4  5.  1  5 

5.  Bought  a  coat  for  $  17.81  ;  a  vest  for  $3.75  ;  a  pair 
of  pantaloons  for  $2.87  ;  and  a  pair  of  boots  for  $  7.18  ; 
what  was  the  amount  ?  Ans.  31.61 


SECT.  8.]  UNITED    STATES'   MONEY.  43 

6.  Sold  a  load  of  wood  for  seven  dollars  six  cents  ;  five 
bushels  of  corn  for  four  dollars  seventy-five  cents,  and 
seven  bushels  of  potatoes  for  two  dollars  six  cents  ; 
what  was  received  for  the  whole  ?  Ans.  $  13.87. 

SUBTRACTION. 


7. 

8. 

9. 

1O. 

From 
Take 

8.      cts.  m. 

61.585 
1  9.  1  9  7 

S.       cts. 

471.81 
1  5  8.  1  9 

$.       cts.   m. 

156.003 
1  9.  0  0  9 

8.        cts. 

141.70 
90.91 

$42.388  $313.62  $136.994   $50.79 

11.  From  $71.07  take  $5.09.  Ans.  $65.98. 

12.  From  $  100.  take  $  17.17.  Ans.  $82.83. 

13.  Bought  a  horse  for  one  hundred  seventy-five  dollars, 
and  sold   him  for  two  hundred  twenty-nine  dollars  eight 
cents  ;   what  was  gained  by  the  bargain  ? 

Ans.  $  54.08. 

14.  From  one  hundred  dollars,  there  was   paid  to  one 
man   seventeen  dollars   nine   cents,   to   another  twenty- 
three  dollars    eight  cents,    and    to   another   thirty-three 
dollars  twenty-five  cents  ;  how  much  cash  remained  ? 

Ans.  $26.58. 

15.  From  ten  dollars  take  nine  mills.         Ans.  $9.991. 

MULTIPLICATION. 

RULE.  Multiply  the  quantity  ly  the  price,  and  in  the 
answer  point,  of  as  many  figures  for  cents  and  mills,  as 
there  are  in  the  price. 

16.  What  cost  143  barrels  of  flour  at  $  7.25  per  barrel  ? 

OPERATION.  Ans.  1036.75. 

1  43 
7.25 
7  15 
286 
1001 

$1036.75  Ans. 


44  UNITED    STATES'    MONEY.  [SECT.  8. 

17.   What  cost  144  gallons  of  oil  at  $  1.625  a  gallon  ? 
OPERATION.  Ans.  $234.00. 

144 
1.625 

720 

288 
864 
144 


$234.000  Ans. 

18.  What  will  165  gallons  of  molasses  cost  at    $0.27 
a  gallon  ?  Ans.  $  44.55. 

19.  Sold  73  tons  of  timber  at  $  5.68  a  ton  ;    what  was 
the  amount  ?  Ans.  $414.64. 

20.  What  cost  43  rakes  at  $  .17  a  piece  ?     Ans.  $  7.31. 

21.  What  cost  19  bushels  of  salt  at  $  1.625  per  bushel  ? 

Ans.  $30.875. 

22.  What  cost  47  acres  of  land  at  $37.75  per  acre  ? 

Ans.  $  1774,25. 

23.  What  cost  19  dozen  penknives  at  $  .375  a  piece  ? 

Ans.  $85.50. 

24.  What  is   the  value  of   17  chests  of  souchong  tea, 
each  weighing  59  pounds,  at  $  .67  per  pound  ? 

Ans.  $672.01. 

25.  When  19  cords  of  wood  are  sold  at  $5,63  per  cord  ; 
what  is  the  amount  ?  Ans.  $  106.97. 

•26.  A  merchant  sold  18  barrels  of  pork,  each  weighing 
200  pounds,  at  12.  cents  5  mills  a  pound  ;  what  did  he 
receive  ?  Ans.  $  450.00. 

27.  A  farmer  sold  one  lot  of  land,  containing  187  acres, 
at  $37.50  per  acre  ;  another  lot,  containing  89  acres,  at 
$  137.37  per  acre  ;  and  another  lot,  containing  57  acres, 
at  $  89.29  per  acre  ;  what  was  the  amount  received  for 
the  whole?  Ans.  $24327.96. 

DIVISION. 

RULE.  Divide  the  price  by  the  quantity,  or  divide  the 
dollars  and  cents  by  the  number  of  things  either  bought 
or  sold,  and  the  quotient  will  be  the  answer,  which  must  be 
pointed  off  like  the  dividend. 


SECT.  9.]  COMPOUND   ADDITION.  45 

28.  If  59  yards  of  cloth  cost   $90.27,  what  cost   one 
yard?  Ans.  $1.53. 

OPERATION. 

5  9  )  9  0.2  7  (  1.53 
59 


29.  If  89  acres  of  land   cost  $  12225.93,    what  is  the 
value  of  one  acre  ?  Ans.  $  137.37. 

30.  When  19  yards  of  cloth  are  sold  for  $  106.97,  what 
should  be  paid  for  one  yard  ?  Ans.  $5.63. 

31.  Gave  $  22.50  for   18  barrels  of   apples  ;  what  was 
paid  for  1  barrel  ?    For  5  barrels  ?    For  10  barrels  ? 

Ans.  $  20.00  for  all. 

32.  Bought  153  pounds  of  tea  for  $90.27  ;   what  was  it 
per  pound  ?  Ans.  $  0.59. 

33.  A  merchant  purchased  a  bale  of  cloth  containing  73 
yards,  for  $414.64  ;  what  was  the  cost-  of  one  yard  ? 

Ans.  $5.68. 


Section  9. 

COMPOUND    ADDITION. 

COMPOUND  ADDITION  is  the  adding  together  of  two  or 
more  numbers  of  different  denominations. 

1.  Paid  a  London  tailor  £7.  13s.  6d.  2qr.  for  a  coat, 
£2.  17s.  9d.  Iqr.  for  a  vest,  £3.  8s.  3d.  3qr.  for  panta 
loons,  and  ,£9.  lls.  8d.  3qr.  for  a  surtout  ;  what  was 
the  amount  of  the  bill  ?  Ans.  £23.  lls.  4d.  Iqr. 

OPERATION.  The  sum  of  the  farthings  in  the 

£••      s-      d.    qr.         right  hand  column  is  9  farthings, 

1362         equal  to  2d.    1  qr.  ;    we   write   the 

21791         farthings   under    the    column    far- 

3        833         things,  and  carry  the  2d.  to  the  col- 

91183         umn  of  pence,  the  sum  of  which  is 

2~3     \~i     4     i         28d.  equal  to  2s.  4d.  ;  we  write  the 


46  COMPOUND   ADDITION.  [SECT.  9. 

4d.  under  its  proper  column,  and  add  the  2s.  to  the 
column  of  shillings,  the  sum  of  which  is  71s.  equal  to 
£3.  lls.  ;  having  written  the  11s.,  we  add  the  £3  to  its 
column,  and  find  the  sum  of  which  to  be  £23.  From 
the  above  process,  we  induce  the  following 

RULE. 

Write  all  the  given  numbers  of  the  same  denomination 
under  each  other;  then  add  the  numbers  of  the  lowest  denom 
ination  together,  and  divide  their  sum  by  so  many  as  make 
one  of  the  next  higher  denomination ;  set  the  remainder  un 
der  its  column,  and  add  the  quotient  to  the  next  column ; 
which  add  together  and  divide  as  before;  thus  proceed  to 
the  last  denomination,  under  which  place  its  whole  sum. 

2.  What  is  the  sum  of  £6.  19s.  lid.  3qr.,   £9.  6s.  3d. 
3qr.,    £13.  18s.  3d.  Iqr.,  and  £67.  Os.  8d.  Iqr.  ? 

Ans.  £97.  5s.  3d.  Oqr. 

TROY  WEIGHT. 
3.  4. 

Ibs.        oz.        dwt.        gr.  Ibs.         oz.        dwt.        gr. 

15     1  1     19     22  10101010 

71101317  81111923 

65        9     17     14  47        7        819 

73111313  16       91014 

14        8        9        9  3310        921 


242        4     14        3 

APOTHECARIES'  WEIGHT. 

5.  6. 

ft-         S-     3-   9-      g^  ft.      §.    5-  9-      gr. 

81116119  3596219 

75     10     7     2     13  7111111 

14        97112  3733212 

37        81111  1447113 

611132        3  7556117 


272       4     3     0     18 


SECT.  9.]  COMPOUND    ADDITION.  47 

AVOIRDUPOIS  WEIGHT. 

7.  8. 

Ton.     cwt.     qr.      Ib.        oz.        dr.  Ton.      cwt.    qr.       Ib.        oz.        dr. 

71  19  3  27  14  13    14  13  2  15  15  15 


14 
39 
15 
61 

13 
9 
17 
16 

1 
3 
3 
3 

1 

1 
1 
1 

1    1 
3 
6    1 
3 

3 
9 

0 

7 

12 
9 
14 

8 

13    1 
46    1 
14    1 
111 

731311 
631113 
5276 
7    3    16    15 

13 
10 
9 
1  1 

203 

17 

3    27 
9. 

8      8 

LONG  MEASURE. 

10. 

deg. 

m. 

fur. 

rd. 

ft.           in.           m.      fur.       rd. 

yd.     ft. 

in. 

18 

19 

7 

15 

1 

1 

1       1 

2 

7     35 

5    2 

1  1 

61 

47 

6 

39 

1 

0     1 

1       1 

3 

6     15 

3     1 

10 

78 

32 

5 

14 

9 

9       1 

6 

1     17 

1     2 

5 

17 

59 

7 

36 

1 

6     1 

0       1 

3 

4     13 

2     1 

9 

28 

56 

1 

30 

1 

6 

1       1 

7 

7     36 

5    2 

7 

205 

8 

1 

17 

1 

5 

2 

LAND 

OR  SQUARE  MEASURE. 

11. 

12. 

A. 

R.      p. 

ft. 

in. 

A. 

R. 

p.       yd.      ft. 

in. 

67 

3    39 

2 

72 

1 

43 

43 

1 

15    3 

0    8 

1  7 

78 

3    1 

4 

260 

1 

16 

16 

3 

39    1 

9    7    1 

41 

14 

2    3 

1 

1 

67 

1 

35 

47 

1 

16    2 

7    5 

79 

67 

1    1 

7 

1 

76 

1 

31 

38 

3 

1  7    1 

8    8 

1  7 

49 

3    3 

1 

69 

1 

17 

15 

1 

32    1 

1    1    1 

17 

278 

3    1 

5 

1 

31 

1 

02 

CLOTH 

MEASURE. 

13. 

14. 

j4 

qr. 

na.    in. 

E.  E.     qr.    na. 

in. 

5 

3 

3    2 

16 

3    2 

1 

7 

1 

1     2 

71 

1     1 

2 

8 

3 

3     1 

13 

3    2 

1 

9 

1 

2    2 

47 

3    2 

2 

4 

3 

3    2 

39 

2    3 

2 

36  3  0  0 


48  COMPOUND    ADDITION.  [SECT.  9. 


SOLID  MEASURE. 
15.                                     16. 

Ton.        ft.             in.                       Cord.           ft.              in. 

1  7     39     1371             14     116     1169 
61     17     1711             67     113     1711 
47161666             96127        969 
71381711             19        98     1376 
47171617             14        371414 

246 

1 

I     1164 
WINE  MEASURE. 

17. 

18. 

Tun. 

hhd. 

gal. 

qt. 

pt. 

Tun. 

hhd. 

gal. 

qt. 

pt 

61 

1 

62 

3 

1 

14 

3 

18 

3 

0 

71 

3 

14 

1 

1 

81 

1 

60 

3 

1 

60 

0 

17 

3 

0 

17 

3 

61 

3 

0 

14 

1 

51 

1 

1 

61 

3 

57 

3 

1 

57 

3 

14 

3 

1 

17 

1 

17 

1 

0 

2 

65 

2 

35 

1 

0 

ALE 

AND 

BEER  MEASURE: 

19 

2O. 

Tun. 

hhd. 

gal. 

qt 

pt. 

Tun. 

hhd. 

gal. 

qt. 

pt. 

15 

3 

50 

3 

1 

67 

1 

51 

1 

0 

67 

3 

17 

3 

1 

15 

3 

16 

3 

1 

17 

1 

44 

1 

0 

44 

1 

45 

1 

1 

71 

3 

12 

3 

1 

15 

2 

12 

2 

1 

81 

1 

18 

1 

0 

67 

3 

35 

1 

0 

254 

1 

36 

0 

1 

DRY 

MEASURE. 

21 

22. 

Ch. 

bu. 

pk.     qt. 

pt. 

Ch. 

bu. 

Pk. 

qt. 

pt. 

15 

35 

3 

7 

1 

71 

17 

1 

1 

1 

61 

16 

3 

6 

1 

16 

31 

3 

3 

0 

51 

30 

1 

6 

0 

41 

14 

3 

1 

1 

42 

1  7 

2 

2 

1 

7  1 

1  7 

1 

0 

1 

14 

14 

1 

4 

1 

10 

10 

2 

3 

0 

1 

86 

7 

1 

1 

0 

SECT.  9.]  COMPOUND    ADDITION.  49 


TIME. 


y. 

d. 

23. 

h.         m. 

s. 

M 

r. 

d. 

24. 

h.         m.           3. 

57     3 

00 

23 

59 

17 

1 

5 

6 

23 

15 

17 

47     1 

69 

15 

17 

38 

6 

1 

5 

15 

27 

18 

29     3 

64 

23 

42 

17 

7 

1 

6 

21 

57 

58 

18     1 

78 

16 

38 

47 

1 

8 

5 

19 

39 

49 

49     3 

17 

20 

52 

57 

87 

6 

19 

18 

57 

203    237 

4 

30 

56 

CIRCULAR 

MOTION. 

25. 

26. 

s. 

0 

1 

u 

S. 

0                 1 

a 

1 

1 

28 

56 

58 

6 

1 

7 

17 

18 

1 

0 

2 

1 

5  1 

37 

7 

09 

19 

51 

8 

1 

3 

39 

57 

8 

1 

8 

57 

45 

8 

1 

9 

38 

49 

4 

1 

7 

16 

39 

7 

1 

7 

47 

48 

7 

27 

38 

48 

1 

1 

1 

1 

55 

09 

MEASURING 

DISTANCES. 

27. 

28. 

m. 

fur.    ch.    p. 

i. 

m. 

fur. 

ch. 

p- 

i. 

1 

7 

5 

8 

3 

24 

14 

7 

9 

3 

21 

1 

6 

;} 

7 

1 

21 

37 

1 

0 

3 

16 

4 

7 

7 

9 

3 

19 

17 

7 

'8 

3 

17 

1 

9 

0 

6 

1 

16 

61 

6 

5 

3 

16 

31 

7 

1 

0 

20 

47 

1 

1 

0 

23 

133     7     4    0     00 


5f  COMPOUND    SUBTRACTION.  [SECT.  10. 

Section  1O. 

COMPOUND    SUBTRACTION. 

COMPOUND  SUBTRACTION  teaches  to  find  the  difference 
between  two  numbers  of  different  denominations. 

1.  A  bill  on  the  bank  of  England  for  £713.  17s.  lid.  3qr. 
was  sold  for  £705.  IGs.  lOd.  Iqr.  ;  what  was  the  sum 
gained  ?  Ans.  £51.  18s.  lOd.  2qr. 

OPERATION.  In  performing  this  ques- 

,%     ,dn     q/'          tlon>  we   cannot  take  3qr. 

,S  i?  Q    fr°m  ^  but  we  can  i°r- 

17     11     6  gimp]e  numbers, 


Ans.  5118102  1  penny  •=.  4qr.,  which  we 

add  to  the  Iqr.  =  5qr.  Take 

3qr.  from  5qr.,  and  2qr.  remain,  which  we  write  under 
the  column  of  farthings  ;  and,  as  in  simple  numbers,  we 
carry  one  to  the  next  lower  number  before  subtracting. 
And  Id.  carried  to  lid.  is  12d.  ;  but,  as  we  cannot  take 
12d.  from  lOd.,  we  must  again  borrow  Is.  from  the  16s. 
=z  12d.  and  add  it  to  the  lOd.  —  22d.  Then  take  12d. 
from  22d.  =  lOd.,  which  we  set  down  and  carry  one,  as 
before,  and  so  on  till  the  whole  be  subtracted.  Hence 
the  following 

RULE. 

Write  those  numbers  under  each  other,  which  are  of  the 
same  denomination,  the  less  compound  number  under  the 
greater.  Begin  with  the  lowest  denomination,  and  subtract 
each  lower  number  from  the  one  above  it,  and  write  the  dif 
ference  underneath.  If  any  lower  number  be  larger  than 
the  upper,  suppose  as  many  to  be  added  to  the  upper  num 
ber  as  would  make  one  of  the  next  higher  denomination,  then 
subtract  the  lower  figure,  remembering  to  carry  one  to  the 
next  lower  number  before  subtracting  it  ;  and  proceed  thus, 
till  all  the  numbers  are  subtracted. 

2.  From  £87.  11s.  9d.  3qr.  take  £41.  5s.  6d.  Iqr. 

Ans.  £46.  6s.  3d.  2qr. 


SECT.  10.]  COMPOUND    SUBTRACTION.  51 


TROY  WEIGHT. 

3.  4. 

lb.         oz.       dwt.       gr.  Ib.        oz.      dwt.       gr. 

15   3  12  14  711  1   3  17 

9  11  17  21  19  3  18  19 


5   3  14  17 

APOTHECARIES'  WEIGHT. 

5.  6. 

ft-     §•    5-    9-     gr.  ft-      §•    3-    9-     gr. 

1571215  16163117 

1197119  9771218 


3 

9 

2    0     16 

AVOIRDUPOIS  WEIGHT. 
7. 

8. 

T. 

cwt.     qr. 

lb. 

oz. 

dr. 

T. 

cwt. 

qr.    lb.       oz. 

dr. 

117 

1 

6 

1 

13 

0 

14 

11 

1 

0     1       1 

13 

19 

1 

7 

3 

27 

1 

15 

9 

18 

3     1     13 

15 

97 

18 

1 

13 

14 

15 

CLOTH  MEASURE. 

9. 

10. 

yd.      qr 

.    na. 

in. 

E.  E.      qr.    na.    in. 

1 

5 

1 

1 

2 

1 

71 

2    2     1 

9 

3 

3 

1 

19 

302 

5 

1 

2 

1 

LONG  MEASURE. 

11. 

12. 

deg. 

m 

fur. 

rd.      yd. 

ft. 

in. 

deg. 

m. 

fur.    rd.      ft. 

in. 

97 

3 

7 

31 

1 

1 

3 

18 

19 

1     1        3 

7 

19 

17 

1 

39 

1 

2 

7 

9 

28 

7     1     16 

9 

77 

56 

I 

31 

5 

0 

2 

52  COMPOUND   SUBTRACTION.  [SECT.  10. 

LAND  OR  SQUARE  MEASURE. 


13. 

A.         R.       p.            ft.             in.            A.         R. 

116     1     13     100     113     139     1 
87317    200     117        973 

14. 

p.         yd.       ft.      in. 

1718130 
18    30     1     31 

28     1     35     172       32 

SOLID  MEASURE. 

15. 

16. 

Tons.         ft.              in. 

171     30     1000 
98     37     1234 

Cord. 

571 

199 

ft.            in. 

18     1234 
19     1279 

72    32     1494 

WINE 

MEASURE. 

17. 

18. 

Tun.      hhd.     gal.    qt.     pt.     gl. 

171     3       8111 
99     1     19     3     1     3 

Tun.     hhd.  gal.    qt.     pt.    gi. 

71     1     1     1     1     1 
933313 

72     1     51     1     1     2 

ALE  AND  BEER  MEASURE. 

19. 

20. 

Tun.    hhd.     gal.      qt.    pt. 

1511710 
9    3     19    3     1 

Tun. 

79 
19 

hhd.      gal.    pt.     qt. 

2       220 
3     13    3     1 

5     1     51     1     1 

DRY 

MEASURE. 

21. 

22. 

Ch.            bu.    pk.    qt.     pt. 

716       1210 
19       9311 

Ch. 

73 

19 

bu.      pk.     qt.    pt. 

13     3    0     1 
18     I     3     1 

696    27    2     7     1 


SECT.  11.]  COMPOUND    ADDITION.  53 

TIME. 
23.  24. 

y.  d.          h.  ma.  w.        d.          h.         m.        a. 

375   151317   5    141    3   415 
199  137  15   1  39     9  6  17  37  48 

175  242  22  15  26 

CIRCULAR  MOTION. 
25.  26. 


S. 


o          i  n  .         o 


11        71315  1233739 

9291736  9153847 


1        7     55    39 

MEASURING  DISTANCES. 
27.  28. 

M.      fur.    ch.     p.        L  M.     fur.    ch.    p.        1. 

2135217  3171119 

9    5    8     1     20  18     1     7     3     23 


11     5    7    0    22 


Section  11. 

EXERCISES  IN  COMPOUND  ADDITION  AND  SUBTRACTION. 

1.  What  is  the    amount  of  the   following  quantities   of 
gold;    41b.  8oz.   13dwt.  8gr.,  51b.   lloz.   19dwt.  23gr., 
81b.  Ooz.  17dwt.  15gr.,  and  181b.  9oz.  14dwt.  lOgr.  ? 

Ans.  371b.  7oz.  5dwt.  8gr. 

2.  An  apothecary  would  mix  71b.  3§.  23.  29.   Igr.  of 
rhubarb,  21b.   10§.  03.   19.   13gr.  of  cantharides,  and 
2ft>.  35.  73.  29.   17gr.  of  opium  ;   what  is  the  amount 
of  the  compound  ?  Ans.  12fc.  5§ . 33.  09.  llgr. 

3.  Add  together  17T.  llcwt.  3qr.  lllb.  I2oz.,  11T.  17cwt. 
Iqr.    191b.    lloz.,    53T.    19cwt.    Iqr.    171b.    Soz.,  27T. 
19cwt.  3qr.  181b.  9oz.,  and  16T.  3cwt.  3qr.  Olb.  13oz. 

Ans.  127T.  12cwt.  Iqr.  121b.  5oz. 

E* 


54  COMPOUND   SUBTRACTION.  [SECT.  11. 

4.  A  merchant  has  3  pieces  of  cloth  ;  the  first  contains 
37yd.  3qr.  Una.,  the  second  18yd.  Iqr.  3na. ,  and  the  third 
31yd.  Iqr.  2na.  ;   what  is  the  whole  quantity  ? 

Ans.  87yd.  3qr.  Ona. 

5.  Sold  3  loads  of  hay  ;    the  first  weighed  2T.   13cwt. 
Iqr.  I71b.,  the  second  3T.  271b.,   and  the  third  IT.  3qr. 
lllb.  ;   what  did  they  all  weigh  ? 

Ans.  6T.  14cwt.  Iqr.  271b. 

6.  What  is  the  sum  of  the  following   distances  ;     16m. 
7fur.    ISr.    14ft.    llin.,   19m.   Ifur.   13r.   l(Jft.  9in.,  97m. 
3fur.  27r.  13ft,  3in.,  and  47m.  5fur.  37r.  13ft.  lUin  ? 

Ans.  181m.  2fur.  18r.  9ft.  3in. 

7.  A  gentleman  has  three   farms,  the  first  contains  169A. 
3R.  15p.  227ft.,  the  second  187A.  1R.  lop.   165ft.,  and 
the  third   217A.   2R.    28p.   165ft.  ;    what    is   the  whole 
quantity  ?  Ans.  574A.  3R.  20p.  12|ft. 

8.  There  are  3  piles  of  wood,  the  first  contains  18cords, 
116ft.  lOOOin.,  the  second   17cords,    lllft.   1600in.,   and 
the  third  21cords,  109ft.  1716in.  ;  how  much  in  all  ? 

Ans.  58cords,  82ft.  860in._ 

9.  John  Thomson  has  four  casks  of  molasses,  the  first 
contains   167gal.  3qt.  Ipt.,  the  second   I86gal.  Iqt.  Ipt., 
the  third   108gal.  2qt.    Ipt.,   and  the  fourth  123gal.  3qt. 
Opt.  ;   how  much  is  the  whole  quantity  ? 

Ans.  536gal.  2qt.  Ipt. 

10.  Add  together  17bu.  Ipk.  7qt.  Ipt.,  18bu.  3pk.  2qt., 
19bu.  Ipk.  3qt.  Ipt.,  and  51bu.  3pk.  Oqt.  Ipt. 

Ans.  107bu.  Ipk.  5qt.  Ipt. 

11.  James  is   13y.  4m.   13da.  old,   Samuel  is  12y.  llm. 
23da.,  and  Daniel  is  I8y.  9m.  29da.  ;   what  is  the  sum 
of  their  united  ages  ?  Ans.  45y.  2mo.  5da. 

12.  Add    together    18y.    345da.    13h.    37m.    15s.,    87y. 
169da.  I2h.  16m.  28s.,  316y.  144da.  20h.  53m.  18s.,  and 
13y.  360da.  21h.  57m.  15s. 

Ans.  436y.  290da.  20h.  44m.  16s. 

13.  Venus  is  3S.  18°.  45'.  15".  east  of  the  sun,  Mars  is 
7S.   15°.  36'.    18".    east   of   Venus,   and    Jupiter  is  5S. 
21°.  38'.  27".  east  of  Mars  ;  how  far  is  Jupiter  east  of 
the  sun  ?  Ans.  4S.  26°. 

14.  A  merchant  owes  a  debt  in  London    amounting  to 
£7671,  what  remains  due  after  he  has  paid  £1728.  17s. 
9d.  ?  Ans.  £5942.  2s.  3d. 


SECT.  11.]  COMPOUND    SUBTRACTION.  55 

15.  From  731b.  of   silver  there  was    made    261b.    lloz. 
13dwt.  14gr.  of  plate  ;   what  quantity  remained  : 

Ans.  461b.  Ooz.  6dwt.  lOgr. 

16.  From  71  fc.  8§.   13.   19.   14gr.   take  7ft>.  9§.   13. 
13.  ITgr.  Ans.  63ft>.  10§.  73.  29.  17gr. 

17.  From23T.  13cwt.  take  10T.  17cwt.  191b.  14oz. 

Ans.  17T.  locwt.  3qr.  81b.  2oz. 

18.  From  76yd.  take  18yd.  3qr.  2na. 

Ans.  57yd.  Oqr.  2na. 

19.  From  20m.  take  3rn.  4fur.  18r.  13ft.  8in. 

Ans.  16m.  3fur.  21r.  2ft.  lOin. 

20.  From  144A.  3R.  take  ISA.  1R.  17p.  200ft.  lOOin. 

Ans.  126A.  1R.  22p.  71ft.  80in. 

21.  From  1'8  cords  take  3  cords  100ft.  lOOOin. 

Ans.  14  cords.  27ft.  728in. 

22.  From  17T.  take  5T.  18ft.  ?65in. 

Ans.  11T.  21ft.  963in. 

23.  From  IGOgal.  take  76gal.  3qt.  Ipt. 

Ans.  92gal.  Oqt.  Ipt. 

24.  From  17Ch.  18bu.  take  5Ch.  20bu.  Ipk.  7qt. 

Ans.  llCh.  33bu.  2pk.  Iqt. 

25.  From  83y.  take  47y.  lOmo.  27da.  18h.  50m.  14s. 

Ans.  35y.  Imo.  2da.  5h.  9m.  46s. 

26.  From  US.  15°.  36'.  15".  take  5S.  18°.  50'.  18". 

Ans.  5S.  26°.  45'.  57". 

27.  A  carpenter  sent  two  of  his  apprentices  to  ascertain 
the  length   of  a  certain  fence.     The   first  stated  it  was 
17r.  16ft.  11  in.,  the  second  said   it   was    18r.  5in.     The 
carpenter  finding  a  discrepancy  in  their  statements,   and 
fearing  they  might  both  be  wrong,  ascertained  the  true 
length  himself,  which  was  17r.  5yd.  1ft.  llin.  ;  how  much 
did  each  differ  from  the  other  ?  Ans. 

28.  From  a  mass  of  silver,  weighing  1061b.,  a  goldsmith 
made   36  spoons,  weighing  51b.    lloz.    12dwt.    15gr.,  a 
tankard,  31b.  Ooz.  13dwt.  14gr.,  a  vase,  71b.  lloz.  14dwt. 
2ogr.  ;   how  much  unwrought  silver  remains  ? 

Ans.  881b.  lloz.  ISdwt.  20gr. 

29.  From  a  piece  of  cloth,  containing  17yd.  3qr.,  there 
were  taken  two  garments,  the  first  measuring  3yd.  3qr. 
2na.,  the  second  4yd.  Iqr.  3na.  ;   how  much  remained  ? 

Ans.  9yd.  Iqr.  3na. 

30.  The  longitude  of  a  certain  star  is  3S.  18°.  14'.  35"., 


56  REDUCTION.  [SECT.  12. 

and  the  longitude  of  Jupiter  is  US.  25°.  30'.  50".  ; 
how  far  will  Jupiter  have  to  move  in  his  orbit  to  be  in 
the  same  longitude  of  the  star  ? 

Ans.  3S.  22°.  43'.  45". 


Section  12. 

REDUCTION. 

MENTAL  OPERATIONS. 

1.  In  2  pence  how  many  farthings  ?    In  4  pence  ?    In  5 
pence  ?     In  7  pence  ?     In  8  pence  ?     In  10  pence  ? 

2.  Plow  many  pence  in  8  farthings  ?    In   12  farthings  ? 
In  16  farthings  ?    In  24  farthings  ?     In  36  farthings  ? 

3.  In  2  shillings  how  many  pence  ?    In  4  shillings  ?    In  5 
shillings  ?     In  6  shillings  ?    In  7  shillings  ? 

4.  In  4  yards  how  many  quarters  ?    In  5  yards  ?    In  6 
yards  ?    In  7  yards  ?     In  8  yards  ?    In  9  yards  ? 

5.  In  8  quarters  how  many  yards  ?     In  12  quarters  ?    In 
16  quarters  ?     In  24  quarters  ?    In  32  quarters  ? 

6.  In   3   feet  how  many  inches  ?     In  5  feet  ?    In  7  feet  ? 
In  8  feet  ?    In  9  feet  ?     In  10  feet  ?     In  12  feet  ? 

*7.  In  36  inches  how  many  feet  ?     In  48  inches  ?     In  60 
inches  ?    In  72  inches  ?    In  96  inches  ?     In  144  inches  ? 

8.  In  6  feet  how  many  yards  ?     In  9  feet  ?     In  12  feet  ? 
In  21  feet  ?    In  24  feet  ?     In  30  feet  ?     In  36  feet  ? 

9.  In  4  yards  how  many  feet  ?    In  3  yards  ?    In  7  yards  ? 
In  9  yards  ?    In  10  yards  ?     In  1 1  yards  ?    In  12  yards  ? 

10.  In  2   acres  how  many  roods  ?     In  3  acres  ?     In  4 
acres  ?     In  6  acres  ?    In  7  acres  ?     In  10  acres  ? 

11.  In  12  roods  how  many  acres  ?     In  8  roods  ?     In  16 
roods  ?    In  20  roods  ?     In  32  roods  ?     In  36  roods  ? 

12.  How  many  furlongs  in  2  miles  ?     In  3  miles  ?     In  6 
miles  ?     In  7  miles  ?     In  8  miles  ?     In  10  miles  ? 

13.  In  12  furlongs  how  many  miles  ?    In  16  furlongs  ?    In 
40  furlongs  ?    In  44  furlongs  ?     In  96  furlongs  ? 

14.  In  5  dimes  how  many   cents  ?     In  6  dimes  ?     In  8 
dimes  ?    In  9  dimes  ?    In  10  dimes  ?    In  12  dimes  ? 


SECT.  12.]  REDUCTION.  57 

15.  How  many  dimes  in  20  cents  ?    In  30  cents  ?    In  40 
cents  ?     In  80  cents  ?     In  90  cents  ?     In  100  cents  ? 

16.  How  many  square  feet  in  1  yard  ?    In  2  yards  ?    In 
3  yards  ?     In  5  yards  ?    In  7  yards  ?    In  8  yards  ? 

17.  In  9  square  feet  how  many  square  yards  ?     In  27 
feet  ?    In  36  feet  ?  In  54  feet  ?  In  63  feet  ?    In  108  feet  ? 

18.  In  1  gallon  how  many  quarts  ?    In  3  gallons  ?    In  5 
gallons  ?    In  7  gallons  ?     In  8  gallons  ?    In  9  gallons  ? 

19.  How  many  gallons  in  4  quarts  ?    In  8  quarts  ?    In  16 
quarts  ?    In  24  quarts  ?     In  32  quarts  ?     In  40  quarts  ? 

20.  How  many  days  in  2  weeks  ?    In  4  weeks  ?    In  5 
weeks  ?     In  7  weeks  ?     In  9  weeks  ?     In  10  weeks  ? 

21.  In  14  days  how  many  weeks  ?    In  21  days  ?    In  28 
days  ?    In  35  days  ?    In  42  days  ?    In  56  days  ? 

22.  How  many  pecks  in  1  bushel  ?    In  3  bushels  ?    In  4 
bushels  ?    In  6  bushels  ?    In  7  bushels  ?    In  9  bushels  ? 

23.  In  8  pecks  how  many  bushels  ?    In  12  pecks  ?    In  16 
pecks  ?    In  24  pecks  ?    In  32  pecks  ?    In  40  pecks  ? 

24.  If  in  1  pound  of  gold  there  are  12  ounces,  how  many 
ounces  in  3  pounds  ?    In  4  pounds  ?    In  6  pounds  ? 

25.  In  24  ounces  how  many  pounds  ?    In  36  ounces  ? 
In  40  ounces  ?    In  60  ounces  ?     In  84  ounces  ? 

26.  In  24  pence  how  many  shillings  ?    In  36  pence  ?    In 
48  pence  ?    In  60  pence  ?    In  72  pencfe  ?    In  144  pence  ? 

The  student  will  now  perceive,  th-{tfj^e,pbject  of 

REDUCTION  is  the  changing  of  numl^feof  one  denomi 
nation  to  another  without  losing  their  value. 

It  consists  of  two  parts,  Descending  and  Ascending. 
The  former  is  performed  by  Mujytiplication,  and  the  latter 
by  Division. 

Reduction  Descending  teaches  to  bring  numbers  of  a 
higher  denomination  to  a  lower  ;  as,  to  bring  pounds  into 
shillings,  or  tons  into  hundred-weights. 

Reduction  Ascending  teaches  to  bring  numbers  of  a 
lower  denomination  into  a  higher  ;  as,  to  bring  farthings 
into  pence,  or  shillings  into  pounds. 


OPERATION. 


58  REDUCTION.  [SECT.  13. 

Section  13. 

REDUCTION    DESCENDING. 

1.  In  16cwt.  3qr.  181b.  how  many  pounds  ? 

Ans.  18941b. 

In  this  question,  we  multiply  the 
I  X  16cwt.  by  4,  because  it  takes  4  quar 
ters  to  make  one  hundred  weight  ;  and 
to  this  product  we  add  the  3qr.  in  the 
question.  Then  we  multiply  by  28, 
because  it  takes  28  pounds  to  make  one 
quarter,  and  to  the  product  we  add  the 
18  pounds  in  the  question,  and  our 
work  is  done. 

From  the  above  illustration,  we  de 
duce  the  following 


RULE. 

Multiply  the  highest  denomination  given  by  so  many  of 
the  next  /ess,  as  will  make  one  of  that  greater ;  and  so  pro 
ceed  until  it  is  brought  to  the  denomination  required,  ob 
serving  to  bring  in  the  lower  denominations  to  their  respec 
tive  places. 

NOTE  1.  To  multiply  by  a  £,  we  divide  the  multiplicand  by  2;  and 
to  multiply  by  a  |,  we  divide  by  4. 

NOTE  2.  The  answers  to  Reduction  Descending  will  be  found  in 
the  questions  of  Reduction  Ascending. 

2.  In  £379  how  many  farthings  ? 

3.  In  £46.  18s.  5d.  how  many  pence  ? 

4.  How  many  grains  Troy  in  371b. 

5.  In  171b.  of  calomel  how  many  grains  ? 

6.  In  15  tons  how  many  ounces  ? 

7.  In  17cwt.  3qr.  191b.  how  many  pounds  ? 

8.  How  many  quarters  in  144  yards  ? 

9.  How  many  nails  in  57  Ells  English  ? 


SECT.  14.]  REDUCTION  59 

10.  How  many  rods  in  97  miles  ? 

11.  How  many  inches  in  7  furlongs  ? 

12.  In  95,000,000  miles  how  many  inches  ? 

13.  In  48deg.  18m.  7fur.  18r.  how  many  feet  ? 

14.  How  many  square  feet  in  76  acres  ? 

15.  How  many  square  yards  in  144  acres  ? 

16.  How  many  square  inches  in  25  square  miles  ? 

17.  How  many  square  feet  in  7A.  3R.  16p.  218ft  ? 

18.  In  15  tons  of  timber  how  many  cubic  inches  ? 

19.  How  many  cubic  inches  in  19  cords,  116  feet  ? 

20.  In  7  hogsheads  of  wine  how  many  pints  ? 

21.  In  5hhd.  17gal.  3qt.  how  many  quarts  ? 

22.  In  17hhd.  of  beer  how  many  pints  ? 

23.  How  many  pints  in  57  bushels  ? 

24.  How  many  quarts  in  15Ch.  16bu.  3pk.  ? 

25.  In  57  days  how  many  minutes  ? 

26.  In  365da.  6h.  how  many  seconds  ? 

27.  In  1842  years  (365da.  6h.  each)  how  many  hours? 

28.  In  8S.  14°.  18'.  17".  how  many  seconds  ? 


Section  14. 

REDUCTION   ASCENDING. 

1.  In  18941b.  how  many  hundred  weight  ? 
OPERAT10N.  Ans.  16cwt.  3qr.  181b. 

O  0   X    1    O  O   A    1U  We  firSt  divide  b7  28» 

2 8  )  1_89  4  Ibs.  because  it  takeg  2gfb  ^ 

4)67.  ISlbs.  make    a    quarter    of    a 

F6cwt.  3qr.  181b.  Ans.     hundred   weight.      We 
then  divide    by   4,   be 
cause  it  takes  4  quarters  to  make  one  hundred  weight. 
Hence  the  following 

RULE. 

Divide  the  lowest  denomination  given  Jjy  that  number, 
which  it  takes  of  that  denomination  to  make  one  of  the  next 
higher ;  so  proceed  until  it  is  brought  to  the  denomination 
Acquired. 


60  REDUCTION.  [SECT.  14. 

NOTE  1.  To  divide  by  5£,  or  16£,  reduce  both  divisors  and  divi 
dends  to  halves  by  multiplying  by  2.  To  divide  by  272^,  reduce  the 
number  to  fourths  by  multiplying  by  4.  If  there  be  a  remainder,  it 
will  be  halves  or  fourths,  like  the  dividend. 

NOTE  2.  The  answers  to  Reduction  Ascending  are  the  questions 
in  Reduction  Descending. 

2.  In  363840  farthings  how  many  pounds  ? 

3.  In  11261  pence  how  many  pounds  ? 

4.  In  213120  grains  Troy  how  many  pounds  ? 

5.  In   97920    grains   how   many    pounds,    Apothecaries' 
weight  ? 

6.  In  537600  ounces  how  many  tons  ? 

*7.  How  many  hundred  weight  in  2007  pounds  ? 

8.  How  many  yards  in  576  quarters  ? 

9.  How  many  ells  English  in  1140  nails  ? 

10.  How  many  miles  in  31040  rods  ? 

11.  How  many  furlongs  in  55440  inches  ? 

12.  How  many  miles  in  6,019,200,000,000  inches  ? 

13.  How  many  degrees  in  17714037  feet  ? 

14.  In  3310560  feet  how  many  acres  ? 

15.  How  many  acres  in  696960  square  yards  ? 

16.  How  many  square  miles  in  100362240000  sq.  in.  ? 

17.  How  many  acres  in  342164  square  feet  ? 

18.  How  many  tons  of  timber  in  1036800  cubic  inches  ? 

19.  How  many  cords  of  wood  in  4402944  cubic  inches  ? 

20.  In  3528  pints  of  wine  how  many  hogsheads  ? 

21.  In  1331  quarts  of  wine  how  many  hogsheads  ? 

22.  In  7344  pints  of  beer  how  many  hogsheads  ? 

23.  How  many  bushels  in  3648  pints  ? 
24:.  How  many  chaldrons  in  17816  quarts  ? 
25.  How  many  days  in  82080  minutes  ? 
£6.  How  many  days  in  31557600  seconds  ? 

27.  How  many  years  in  16146972  hours  ? 

28.  In  915497"  how  many  signs  ? 


SECT.  15.]  MISCELLANEOUS.  61 

Section  15. 

MISCELLANEOUS. 

QUESTIONS  TO  EXERCISE  THE  FOREGOING  RULES. 

1.  At  $  5  per  ream,  how  many  reams  can  be  bought  for 
$  175  ?  Ans.  35  reams. 

2.  At  $7.50  per  barrel,  how  many  barrels  of  flour  can 
be  obtained  for  $217.50  ?  Ans.  29  barrels. 

3.  At  8  75  per  ton,  how  many  tons  of  iron  can  be  pur 
chased  for  8  4875  ?  Ans.  65  tons. 

4.  At  84   per  yard,   how  many  yards  of  cloth  can  be 
bought  for  8  1728  ?  Ans.  432  yards. 

5.  If  a  ton  of  coals  cost  8  8.40,  what  cost  one  cwt.  ? 

Ans.  42  cents. 

6.  At  82.40  per  bu.,  what  cost  1  pk.  ?    What  cost  17bu. 
3pk.  Ans.  8  42.60. 

7.  At  8  3.50  per  quintal,  what  cost  37  quintals  ? 

Ans.  8  129.50. 

8.  John  Webster  bought  5cwt.  3qr.  181b.  of  sugar  at  9 
cents  per  lb.,  for  which  he  paid  25  barrels  of  apples  at 
8  1.75  per  barrel  ;  how  much  remains  due  ? 

Ans.  8  15.83. 

9.  If  971b.  of  beef  cost  88.73,  what  cost   lib.  ?    What 
cost  1471b.  ?  Ans.  813.23. 

10.  If  a  man  travel  45  miles  in  9  hours,  how  far  will  he 
travel  in  1  hour  ?    How  far  in  59  hours  ? 

Ans.  295  miles. 

11.  If  a  ton  of  hay  can  be  purchased  for  8  18.40,  what 
will  be  the  price  of  Icwt.  ?    What  of  47cwt.  ? 

Ans.  8  43.24. 

12.  Bought  65  barrels  of  flour  for  8  422.50,  what  cost 
one  barrel  ?    What  cost  15  barrels  ?         Ans.  8  97.50. 

13.  For  45  acres  of  land,  a  farmer  paid  8  2025  ;  what 
cost  one  acre  ?    What  ISO  acres  ?        Ans.  88100.00. 

14.  For  5  pairs  of  gloves,  a  lady  paid  83.45  ;  what  cost 
1  pair  ?    What  cost  11  pairs  ?  Ans.  87.59. 

15.  When  $  1480  are  paid  for  25  acres  of  land,  what 
cost  1  acre  ?    What  cost  1  rod  ?    What  cost  37A.  2R. 
l§p.  Ans.  $2226.66. 

F 


62  MISCELLANEOUS.  [SECT.  15. 

16.  Paid   $  10.08  for  1441b.  of  pepper  ;    what  was  the 
price  of  one  pound  ?    What  cost  3591b.  ? 

Ans.  $25.13. 

17.  Paid    $77.13  for  S571b.  of  rice  ;    what   cost  lib.  ? 
What  cost  3591b.  ?  Ans.  $  32.31. 

18.  J.  Johnson  paid  $  187.53  for  987gal.  of  molasses  ? 
what  cost  Igal.  ?    What  cost  329gal.  ?     Ans.  $62.51. 

19.  For  47  bushels  of  salt,  J.  Ingersoll  paid  $26.32; 
what  cost  1  bushel  ?    What  cost  39  bushels  ? 

Ans.  $21.84. 

20.  If  15  men  can  perform  a  piece  of  work  in  10  days, 
how  long  would  it  take  one  man  to  perform  the  same  la 
bor  ?    How  long  75  men  ?  Ans.  2  days. 

21.  A  certain  field  will  pasture  10  horses  9  weeks  ;  how 
long  will  it  pasture  1  horse  ?    How  long  18  horses  ? 

Ans.  5  weeks. 

22.  If  a  mechanic,  by  laboring  9  hours   per  day,  can 
perform  a  certain  piece  of  work  in  10  days,  how  long 
would  it  take  him  by  laboring  one  hour  per  day  ?    How 
long  by  15  hours  per  day  ?  Ans.  6  days. 

23.  Bought    a    silver   tankard,    weighing   21b.   7oz.   for 
$46.50  ;    what  did  it  cost  per  oz.  ?    How  much  per  Ib.  ? 

Ans.  $  18.00. 

24.  Bought  3T.   Icwt.  181b.  of  leather  at  12  cents  per 
Ib.,  and  sold  it  at  9  cents  per  Ib.  ;  what  did  I  lose  ? 

Ans.  $205.50. 

25.  Phineas  Bailey  has  agreed  to  grade  a  certain  rail 
road  at  $5.75  per  rod  ;   what  will  he  receive  for  grad 
ing  a  road  between  two  cities,  whose  distance  from  each 
other  is  37m.  7fur.  29r.  ?  Ans.  f  69856.75. 

26.  Bought  a  hogshead  of  molasses,  containing  100  gal 
lons,  for  $25  ;   but  15gal.  3qt.  having  leaked  out,  I  sold 
the  remainder  at  12  cents  a  quart  ;  what  did  I  gain  ? 

Ans.  $15.44. 

27.  From   a    large    farm,    containing  765 A.   3R.    14p., 
there  were  sold  144A.  at  $75  per  acre,  and  the  remain 
der  was  sold  at   $  1.67  per  square  rod  ;   what  was  the 
whole  amount  ?  Ans.  $  176954.98. 

28.  Bought    15T.    3cwt.    151b.    of   iron  at  6  cents  per 
pound  ;   sold  6T.  Icwt.  Iqr.  I81b.  at  5  cents  per  Ib.,  and 
the  remainder  at  10  cents  per  Ib.  ;  what  did  I  gain  ? 

Ans.  $678.14, 


SECT.  16.]        COMPOUND    MULTIPLICATION.  63 

29.  John  Smith  has  3  farms,  the  first  contains  89A.  3R. 
39p.  ;  the  second  97A.  1R.  15p.  ;   and  the  third  117A. 
1R.  19p.     He  gave  his  son  175A.  3R.  29p.  and  he  sold 
the  remainder  at  $  1.25  per  square  rod.     What  did  he 
receive  ?  Ans.  $25755.00. 

30.  A  lady  gave  her  daughter  $  10  to  go  a  "  shopping  "; 
having  purchased  2yd.  of  silk,  at  $  1.27  per  yd.,  a  bon 
net  for  $3.75,  3  pairs  of  gloves  at  0.19  a  pair,  and  two 
fans  at  0.37  each,  she  returned  the  remainder  of  the 
money  to  her  mother  ;  what  was  the  sum  ? 

I  Ans.  $2.40. 


Section  16. 

COMPOUND    MULTIPLICATION. 

MENTAL  OPERATIONS. 

1.  If  a  penknife  cost  9d.,  what  will  2  penknives  cost  ? 
What  will  3  ?    What  will  4  ? 

2.  If  a  yard  of  cloth  cost  Is.  6d.,  what  will  2yd.  cost  ? 
4yd.  ?    6  yd.  ?    7yd.  ? 

3.  A  boy  bought  a  top,   for  Is.  2d.  ;  what  will  3  tops 
cost  ?     What  will  5  tops  cost  ? 

4.  If  a   man  walk  7m.  4fur.  in  1   day,   how  far  will  he 
walk  in  2  days  ?    In  3  days  ?     In  5  days  ? 

5.  If  a  man   consume  51b.  6oz.  of  meat  in  1  week,   how 
much  will  he  require  in  3  weeks  ? 

6.  If  a   small   book   cost  9d.,  what  will  2  books  cost  ? 
What  will  4  books  ?    What  will  6  books  ? 

FOR    THE    SLATE. 

1.  If  an  acre  of  land  cost  £  14.  5s.  8d.  2qr.,  what  will 
9  acres  cost  ?  Ans.  £  128.  lls.  4d.  2qr. 


OPERATION. 


In  performing  this  question,  we 
d-    v-  say  9  times  2  farthings  are  18  far 

things  ;    these   farthings,   we    re 
duce  to  pence  by  dividing  them  by 


1281142  4  :  and  we  find  the  result  to  be  4d. 


64  COMPOUND    MULTIPLICATION.        [SECT.  16. 

and  2  farthings  remaining.  We  set  down  the  2  farthings 
and  carry  4  to  the  next  product.  We  then  say  9  times  8 
pence  are  72  pence,  to  these  we  add  the  4  pence,  which 
make  76  pence,  which  we  divide  by  12,  the  number  of 
pence  in  a  shilling,  and  find  the  result  to  be  6  shillings  and 
4  pence,  we  set  down  the  pence  and"  carry  the  6  shillings 
to  the  next  product.  We  then  say  9  times  5  shillings  are 
45  shillings,  to  these  we  add  the  6  shillings,  and  the  sum 
is  51  shillings,  which  are  equal  to  2  pounds  and  11  shil 
lings.  We  set  down  the  11  shillings,  and  carry  the  2 
pounds  to  the  next  product,  and  then  say  9  times  14 
pounds  are  126  pounds,  to  these  we  add  the  2  pounds,  and 
the  sum  is  128  pounds,  which  we  set  down  under  the 
pounds  in  the  multiplicand,  and  the  work  is  finished,  arid 
the  answer  is  £128.  11s.  4d.  2qr.  Hence  we  perceive, 
that  when  the  quantity  is  less  than  12,  we  may  adopt  the 
following 

RULE. 

Multiply  each  denomination  of  the  compound  number^ 
beginning  at  the  lowest,  by  the  multiplier,  and  carry  as  in 
Compound  Addition. 

5. 

£.         s.       d. 

18    15   8} 


2. 

3. 

4. 

£. 

B.      d. 

£. 

s. 

d. 

£. 

8. 

d. 

5 

6   8 

19 

1  1 

7 

25 

17 

1  1 

2 

3 

5 

10134      58149      129       9       7      11214 


6. 

7. 

8. 

Cwt. 

qr.       lb. 

oz. 

Ton. 

cwt. 

qr. 

lb. 

Cpt. 

qr. 

lb. 

oz. 

18 

3    17 

10 

14 

15 

3 

12 

19 

1 

8 

15 

6 

7 

8 

113    1    21    12     103    1100     154    2    15       8 
9.  1O.  11. 

Ibs.        oz.      dr.  M.      fur.    rd.        ft.  Deg.        m.    fur.     rd. 

151413  9771413  1812618 

968 


143       5       5        587    4      8    12        145    32    7    24 


SECT.  1C.]        COMPOUND  MULTIPLICATION.  65 

12.  13. 

Rd.      yd.    ft.     in.  Fur.     rd.          ft.         in. 

23    3    2    9  9    31     16     11 

9  10 


213    209  98       0       4       2 

NOTE.  The  answers  to  the  following  questions  are  found  in  the 
corresponding  numbers  in  Compound  Division. 

14.  What  cost  7  yards  of  cloth  at  18s.  9d.  per  yard  ? 

15.  If  a  man  travel  12m.  3fur.  29rd.  in  one  day,  how  far 
will  he  travel  in  9  days  ? 

16.  If  1  acre   produce  2  tons  13cwt.  191b.  of  hay,  what 
will  8  acres  pr.oduce  ? 

17.  If  a  family  consume  49galls.  3qts.  Ipt.  of  molasses 
in  1  month,  what  quantity  will  be  sufficient  for  one  year  ? 

18.  John  Smith  has  12  silver  spoons,  each  weighing  3oz. 
ITdwt.  14gr.,  what  is  the  weight  of  all  ? 

19.  Samuel    Johnson    bought  7   loads   of  timber,   each 
measuring  7  tons  37ft.  ;   what  was  the  whole  quantity  ? 

20.  If  the  moon  move  in  her  orbit  13°.    11'.  35".  in  1 
day,  how  far  will  she  move  in  10  days  ? 

21.  If  1  dollar  will  purchase  21b.  8§.  73.  19.  lOgr.  of 
ipecacuana,  what  quantity  would  9  dollars  buy  ? 

22.  If   1  dollar  will  buy  2A.  3R.   15p.  30yd.  8ft.   lOOin. 
of  wild   land,  what  quantity  may  be  purchased  for  12 
dollars  ? 

23.  Joseph  Doe  will    cut  2  cords   97ft.  of  wood    in  1 
day  ;  how  much  will  he  cut  in  9  days  ? 

24.  If  1   acre  of  land  produce  3ch.  6bu.  2pk.  7qt.  Ipt. 
of  corn,  what  will  8  acres  produce  ? 

II.  If  the  quantity  be  such  as  may  be  resolved  into 
two  or  more  factors,  that  is,  two  or  more  numbers,  whose 
product  shall  be  equal  to  the  quantity,  the  compound  num 
ber  may  be  multiplied  by  1  of  those  numbers,  and  the 
product  by  the  other,  and  the  last  product  will  be  the  value 
of  the  whole  quantity. 

25.  What  cost  24  yards  of  broadcloth  at  £2.  7s.  lid. 
per  yard  ? 


66  COMPOUND    MULTIPLICATION.       [SECT.  16, 

g       d  In  this  question,  we  find  the  quanti- 

2        711  ty  ^  equal  to  the  product  of  4  and  6, 

.  we  therefore  multiply  the  price  first  by 

4,  and  then  that  product  by  6,  and  the 


last   product  is  the   answer.      Or  we 
might  have  multiplied  first  by  6  and 


_ 

5710        0  then  by  4,  and  the  answer  would  have 

been  the  same. 

26.  What  cost  360  tons  of   iron  at   £17.  16s.  Id.  per 
ton  ? 

£"  *•  d-  In  this  question,  we  find  the  factors 
1716  1  of  360  to  be  6  and  6  and  10,  that  is,  6 
multiplied  by  6  are  36,  and  36  multiplied 


106     16        6  by  10  make  360.     We  then  first  multi- 

6  ply  by  6,  and  then  that  product   by  6, 

*•  4  Q -TQ        Q  and  then  again  the  last  product  by  10. 

I  Q  The  result  wolud  have  been  the  same, 

— — if  we  had  multiplied  by  10  first. 

27.  If  a  man  travel  3m.  7fur.  18rds.  in  one  day,  how  far 
would  he  travel  in  30  days  ? 

28.  If  a  load  of  hay  weigh  2  tons  7cwt.  3qrs.  181b.,  what 
would  be  the  weight  of  84  similar  loads  ? 

29.  When  it  requires  7yds.  3qr.  2na.  of  silk  to  make  a 
lady's  dress,  what  quantity  would  be  sufficient  to  make 
72  similar  dresses  ? 

30.  A  tailor  has  an  order  from  the  navy  agent  to  make 
132  garments  for  seamen  ;  how  much  cloth  will  it  take, 
supposing  each  garment  to  require  3yds.  2qr.  Ina.  ? 

III.  When  the  quantity  is  more  than  12,  and  the  num 
ber  is  such,  that  it  cannot  be  resolved  into  two  or  more 
factors,  the  better  method  is  to  find  the  factors  of  a  num 
ber  nearest  the  given  number,  and  having  multiplied  the 
compound  number  by  one  of  these  factors,  and  the  product 
by  the  other  factor,  then  find  the  value  of  the  remaining 
quantity  and  add  it  to  the  last  product. 

31.  If  1  dollar  will  buy  171bs.  lOoz.  13dr.  of  beef,  how 
much  may  be  bought  for  62  dollars  ? 


SECT.  17.]  COMPOUND    DIVISION.  67 


Ib. 

17 

oz. 

10 

dr. 

13 
5 

lb-       oz-       dr-            As  62  is  not  the 
17     10     13      product  of  any  two 
^      numbers  in  the  mul- 

88 

6 

1 
12 

35        510      tiplication  table,  we 
take  some  conveni 
ent   number   less   than   62,    viz.    60. 
This  we  resolve  into  two  factors  5 
and  12,  and  having  found  the  amount 
of  60  dollars,  we  then  find  the   quan- 

1060 
35 

8 
5 

12 
10 

1095 

14 

6 

tity  2  dollars  will  buy,  and  add  this  amount  to  the  former, 
and  the  sum  is  the  quantity  62  dollars  will  buy. 

32.  What  cost  97  tons  of  lead  at  £2.  17s.  9|d.  per  ton  ? 

33.  If  a  man  travel   17m.  3fur.  19r.  3yd.  2ft.  7in.  in  one 
day,  how  far  would  he  travel  in  38  days  ? 

34.  If  1  acre  will  produce  27bu.  3pk.  6qt.  Ipt.  of  corn, 
what  will  98  acres  produce  ? 

35..  If  it  require  7yd.  oqr.  2na.  to  make  1  cloak,  what 
quantity  would  it  require  to  make  4S  cloaks  ? 

36.  One  ton  of  iron  will  buy  13A.  3R.  14p.  18yd.  7ft. 
76in.  of  land  ;  how  many  acres  will  19  tons  buy  ? 


Section  17. 

COMPOUND    DIVISION. 

MENTAL  OPERATIONS. 

1.  If  2  yards  of  cloth  cost  3s.,  what  will  1  yard  cost  ? 

2.  If  3  barrels  of  apples  cost  5s.,  what  cost  1  barrel  ? 

3.  If  4hhds.  of  lime  cost  15s.,  what  cost  Ihhd.  ? 

4.  Divide  9s.  equally  among  9  boys. 

5.  Divide  lOd.  equally  among  3  girls. 

6.  What  is  a  fourth  part  of  5  gallons  ; 

7.  What  is  a  seventh  part  of  7  gallons  ? 

8.  What  is  a  sixth  part  of  9  gallons  ? 

FOR    THE    SLATE. 

1.  If  9  acres  of  land  cost  £128.  11s.  8d.  2qr.,  what  is 
the  value  of  1  acre  ?  Ans.  £  14.  5s.  8d.  2qr. 


63  COMPOUND    DIVISION.  [SECT.  17. 

OPERATION.  Having  divided  the  pounds  by 

£.          *.     d.    qr.         9,   we  find   the   quotient   to   be 
9)128     11     4     2         £14,     which    we    write    under 
j~4        5     g     ^         £128,  and  to  the  £2.  remain 
ing   (40s.)   we  add   the   11s.  in 

question,  and  their  amount  is  51s.  ;  and  these  51s.  we 
again  divide  by  9,  and  the  quotient  is  5s.,  which  we  write 
under  the  lls.  in  the  question  ;  and  to  the  remainder,  6s., 
which  are  72d.,  we  add  the  4d.  in  the  question,  and  the 
sum  is  76d.  ;  having  again  divided  these  by  9,  we  write 
the  quotient,  8,  under  the  4d.  in  the  question  ;  and  to  the 
remainder,  4d.,  which  is  16qr.,  we  add  the  2qr.  in  the  ques 
tion,  and  the  amount  is  18qr.,  which  we  again  divide  by 
9,  and  find  the  quotient  to  be  2qr.,  which  we  write  under 
the  2qr.  in  the  question.  Thus  we  find  our  answer  to  the 
question  to  be  <£  14.  5s.  8d.  2qr.  Hence  the  following 

RULE. 

I.  Divide  the  highest  denomination  ly  the  quantity ;  and 
if  any  thing  remains,  reduce  it  to  the  next  lower  denomina 
tion,  and  continue  to  divide  until  it  is  reduced  to  the  lowest 
denomination. 

2.  3.  4. 

£.  8.  d.  £.  8.  d.  £.  8.  d. 

2)10     13    4          3)58     14     9          5)129       9       7 
5        68  19117  251711 

5.  6.  7. 

£,.         a.     d.  qr.  Cwt.    qr.    Ib.       oz.  Ton.    cwt.  qr.      Ib. 

6)112  14  4  2  6)113  1  21  12  7)103  11  0   0 


18 

15 

8 

3 

18 

3 

17 

10 

14  15 

3 

12 

8. 

9. 

10. 

Cwt. 

qr,  Ib. 

oz. 

Ib. 

oz. 

dr. 

M.   fur. 

rd. 

ft. 

?) 

154 

2  1 

5 

8 

9)143 

5 

5 

6)587  4 

8 

12 

19 

1 

8 

15 

15 

14 

13 

11. 

12. 

13. 

Deg. 

m. 

fur. 

rd. 

Rd. 

yd. 

ft.  in. 

Fur.   rd 

.  ft 

.  in. 

8) 

145 

32 

7 

24 

9)2 

13 

2 

0  9 

10)98  0 

4 

2 

NOTE.     The  answers  to  the  following  questions   are  found  in  the 
corresponding  numbers  in  Compound  Multiplication. 


SECT.  17.]  COMPOUND    DIVISION.  69 

14.  What  cost  1  yard  of  cloth,  when  7yd.  can  be  bought 
for  JE6.  11s.  3d.  ? 

15.  If  a  man,  in  9  days,  travel  112m.  Ifur.  21rd.,  how 
far  will  he  travel  in  1  day  ? 

16.  If  8  acres  produce  21T.  5cwt.    Iqr.    121b.   of  hay, 
what  will  1  acre  produce  ? 

17.  If  a  family  consume  in  1  year  598gal.  2qt.  of  molas 
ses,  how  much  may  be  necessary  for  1  month  ? 

18.  John  Smith  has  12  silver  spoons,  weighing  31b.  lOoz. 
lldwt.  ;  what  is  the  weight  of  each  spoon  ? 

19.  Samuel  Johnson  bought  7  loads  of  timber,  measuring 
55T.  19ft.  ;  what  was  the  quantity  in  each  load  ? 

20.  If  the  moon,  in  10  days,  move  in  her  orbit  4S.   11°. 
55'.  50".,  how  far  does  she  move  in  1  day  ? 

21.  If  $9  will  buy  241b.  8§.  33.  19.  lOgr.  of  ipecacu 
anha,  how  large  a  quantity  will  $  1  purchase  ? 

22.  When  §  12  will  buy  34A.  OR.  32p.  8yd.  5ft.  48in.  of 
wild  land  ;  how  much  will  $  1  buy  ? 

23.  Joseph  Doe  will  cut  24  cords  105  feet  of  wood  in  9 
days  ;  how  much  will  he  cut  in  1  day  ? 

24.  When  8   acres  of  land   produce  25Ch.   17bu.  3pk. 
4qt.  of  grain  ;  what  will  1  acre  produce  ? 

When  the  quantity  is  a  composite  number,  that  is,  one 
which  is  composed  of  the  product  of  two  or  more  num 
bers,  we  proceed  as  in  the  following  question. 

25.  When  24  yards  of  broadcloth  are  sold  for  £57.  10s. 
Od.,  what  is  the  price  of  1  yard  ?      Ans.  ,£2.  7s.  lid. 

£.        s.         d.  In  this  question,  we  find  the  com- 

6)57     10        0         ponent  parts,  or  factors,  of  24  are 

4~)~9 T~I        8         ^  an(^  ^  ;  that  is,  6  multiplied  by  4 

oo~ = — j-r         produces   24.      We  therefore   first 

divide   the    price   by  one   of  these 

numbers,  and  then  divide  the  quotient  by  the  other.  From 
the  above  process  we  deduce  the  following 

RULE. 

II.  Divide  the  dividend  ly  one  of  the  component  parts, 
and  the  quotient  thence  arising  by  the  other,  and  the  last 
quotient  will  oe  the  answer. 

When  the   quantity  is  such,  that  it  cannot  be  resolved 


70  COMPOUND    DIVISION.  [SECT.  17. 

into  two  or  more  factors,  the  question  must  be  performed 
by  Long  Division,  as  in  the  following  question. 

26.  If  23cwt.  of  iron  cost  £171.  Is.  3d.  what  cost  Icwt.  ? 

OPERATION.  Ans.  £  7.  8s.  9d. 

Q/  -r-r        In  this  question  we   first  divide 

the  p°unds  by  23>  and  obtain  7. for 

the   quotient,  and  .£10  remaining, 
we  reduce  to  shillings   and   annex 

the  Is.  and  again  divide  by  23  and 

23)20  I  (8s.  obtain   8s.  for  the  quotient.      The 

184  remainder,  17s.,  we  reduce  to  pence 

j~7  and  annex  the  3d.  and  again  divide 

j  2  by  23,  and  obtain  9d.  for  the  quo- 

«  oTTTT-^  /r.j  tient.     Thus  we  find  the  answer  to 

23)207(9d.  be  £7.  8s.  9d. 

So  in  similar  cases  we  should  divide  the  highest  de 
nomination  by  the  quantity,  and  if  any  thing  remains, 
reduce  it  to  the  next  lower  denomination  and  continue  to 
divide  until  it  is  reduced  to  the  lowest  denomination. 

27.  If  a  man  travel  117m.  7fur.  20rd.  in  30  days,  how 
far  will  he  travel  in  1  day  ? 

28.  If  84  loads  of  hay  weigh  201  Tons  4cwt.  2qr.  Olb., 
what  will  1  load  weigh  ? 

29.  When  72  ladies  require  567yd.  Oqr.  Ona.  for  their 
dresses,  how  many  yards  will  be  necessary  for  1  lady  ? 

30.  When  132  sailors   require  470yd.   Iqr.   of  cloth  to 
make  their  garments,  how  many  yards  will  be  necessary 
for  1  sailor  ? 

31.  If  $62    will   buy  10951b.  14oz.  6dr.  of  beef,    how 
much  may  be  obtained  for  $  1  ? 

32.  Paid  £280.  5s.  9Jd.  for  97  tons  of  lead  ;   what  did 
it  cost  per  ton  ? 

33.  If  a  man  travel  662m.  4fur.  28rd.  3yd.  2ft.  2in.  in  38 
days,  how  far  will  he  travel  in  1  day  ? 

34.  When    98    acres    produce    2739bu.    Ipk.    5qt.    of 
grain,  what  will  1  acre  produce  ? 

35.  A  tailor  made  48  garments  from  378  yards  of  cloth  ; 
what  quantity  would  it  take  to  make  1  garment  ? 

36.  When  19  tons  of  iron  will  purchase  262A.  3R.  37p. 
25yd.  1ft.  40in.  of  land,  how  much  may  be  obtained  for 
1  ton? 


SECT.  18.] 


BILLS. 


71 


Section  18. 


BILLS. 


Mr.  William  Greenleaf, 


Haverhill,  March  19,  1842. 


86  Shovels, 
90  Spades, 
18  Ploughs, 
Handsaws 
"ammers, 
tfillsaws, 
:.  Iron, 


Bought  of  Moses  Atwood, 
at          $  0.50. 
86. 
11.00. 
3.50. 
62. 
12.12. 


12.00. 


ived  payment, 


1105.02. 


Moses  Atwood. 


Lowell,  June  5,  1842. 
Amos  Dow, 

Bought  of  Lord  &  Greenleaf. 
37  Chests  Green  Tea,  at         $  23  75 

42  "       Black    do.  «  17.'50.' 

43  Casks  Wine,  "  99.00. 
Crates  Liverpool  Ware,      "           175.00. 
bis.  Genessee  Flour,            "  7.00. 


Received  payment, 


$8138.71, 

Lord  &  Greenleaf, 

by  James  Clark. 


72  BILLS.  [SECT.  18. 

Baltimore,  July  19,  1842. 
Mr.  John  Kimball, 

Bought  of  Simon  Grey, 

14  oz.  Gum  Camphor,  at         $  0.63. 

12  "    Laudanum,  "  .88. 

23  "    Gum  Elastic,  "  .62. 

16  "    Emetic  Tartar,  "  1.27. 

17  "    Cantharides,  "  2.25. 

$92.21. 

Received  payment, 

Simon  Grey, 

by  Enoch  Osgood. 


New  York,  May  20,  1842. 
Dr.  John  Smith, 

Bought  of  Somes  &  Gridley, 
82  galls.  Temperance  Wine,         at         $  .75. 
89     "      Port,  do.  "  .92. 

24  pair  Silk  Gloves,  "  .50. 

~!Tl55.38. 

Received  payment, 

Somes  &  Gridley. 


Newburyport,  March  7,  1842. 
Mr.  Levi  Webster, 

Bought  of  James  Frankland, 

6  Ibs.  Chocolate,  at         $  .18. 

12   "     Flour,  "  .20. 

6  pair  Shoes,  "  1.80. 

30  Ibs.  Candles,  "  .26. 

$22.08. 
Received  payment, 

James  Frankland. 


SECT.  18.]  BILLS.  73 

Salem,  May  13,  1842. 
Mr.  Noah  Webster, 

Bought  of  Ayer,  Fitts,  &  Co. 

80  pair  Hose,                              at  $  1.20. 

17     "     Boots,                              "  3.00. 

19     "     Shoes,                              "  1.08 
23     "     Gloves, 


$  184.77. 
Received  payment, 

Ayer,  Fitts,  &  Co. 

by  William  Summers. 


Baltimore,  June  30,  1842. 
Mr.  Samuel  Osgood, 

Bought  of  Stephen  Barnwell, 
27  Young  Readers,  at  $  .20. 

10  Greek  Lexicons,  "  3. 90. 

7  Ainsworth's  Dictionaries,        "  4.75. 

19  Folio  Bibles,  "  2.93 

20  Testaments,  "  .'37. 

~!Tl40.72. 
Received  payment, 

Stephen  Barnwell. 


Philadelphia,  August  1,  1842. 
Mr.  Elias  Smith, 

Bought  of  Timothy  Eaton, 

49  yds.  Calico,  at          $ .30. 

Irish  Linen,  "  2.56. 

140   ps.    Nankin,  "  2.91. 

169  yds.  Pongee  Silk,  "  200 

153     "     Blue        do.  "  i.*37.' 

8  1087.47. 
Received  payment, 

Timothy  Eaton. 


74  FRACTIONS.  [SECT.  19. 

London,  June  19,  1842. 
Mr.  Edward  Snow  of  Lowell,  U.  S. 

Bought  of  Smith,  Davis,  &.  Co. 
241  yds.  Red  Broadcloth,      at  16s.  4d. 
412     "     Blue          do.  "     8s.  9d. 

510     "     White       do.  "  13s.  5£d. 

424     "     Green       do.  "  14s.  6£d. 

169     "     Black  Velvet,  "   12s.  84d. 

349     "     Black  Kerseymere,  "   17s.  6jd. 
648     "     Carpet,  "   14s.  9^d. 

£1919.  18s.  9id. 
Received  payment, 

Smith,  Davis,  &  Co. 

by  Thomas  Vance. 


Section  19. 

FRACTIONS. 

MENTAL  OPERATIONS. 

The  pupil  must  carefully  commit  all  the  definitions  on 
page  77,  before  he  commences  mental  operations. 

1.  If  an  apple   be  divided  into  two  equal  parts,  one  of 
those  parts  is  called  a  half,  and  is  written  thus,  ^-. 

2.  If  an  apple  be  divided  into  three  equal  parts,  one  of 
those  parts  is  called  a  third,  and  is  written  thus,  •£. 

3.  Two  of  those  parts  are  called  two  thirds,  and  are  writ 
ten  thus,  §. 

4.  If  an  orange  is  divided  into  four  equal  parts,  one  of 
those   parts  is  called  a  quarter,  and  is  written  thus,  {. 
Two  of  those  parts  are  called  two  fourths,  and  are  writ 
ten  thus,  f,  or  thus,  £. 

5.  Three  of  those  parts  are  called  three  quarters,  and 
are  written  thus,  £-. 

6.  One  is  what  part  of  two  ?  Ans.  ^. 

7.  One  is  what  part  of  three  ?  Ans.  £. 


SECT   19.]  FRACTIONS.  75 

8.  One  is  what  part  of  four  ?    Ans.  £.     What  part  of  5  ? 

9.  Two  is  what  part  of  3  ?  Ans.  f . 

10.  What  part  of  5  is  2  ?     Is  3  ?     Is  4  ?     Is  6  ?     Is  7  ? 

11.  What  part  of  7  is  2  ?    Is  3  ?    Is  5  ?    Is  6  ? 

12.  What  part  of  1 1  is  4  ?     Is  5  ?    Is  6  ?    Is  7  ? 

13.  What  part  of  19  is  5  ?    Is  1 1  ?    Is  13  ?    Is  17  ? 

14.  When  corn  is  7  shillings  a  bushel,  what  part  of  a 
bushel  could  you  buy  for  Is.  ?    For  2s.  ?    For  5s.  ? 

15.  When   flour  is  $  9  per  barrel,  what  part  of  a  barrel 
could  be  bought  for  8  2  ?    For  8  3  ?    For  8  7  ? 

16.  If  £  of  a  barrel  of  flour  cost  8  2,  what  will  f  cost  ? 
What  will  f  ?    What  will  |  ?    What  will  £  ? 

17.  If  I-  of  a  cwt.  of  sugar  cost  $  14,  what  will  $  cost  ? 

18.  What  will  f  cost  ?    $  ?    f  ?    £  ?    f  ?    f  ? 

19.  If  T7T  of  a  pound  of  tea  cost  35  cents,  what  will  TJT 

COSt?      T3T?      T5T?      T6f?      T8r?     .W? 

20.  If  T5T  of  a  yard  of  cloth  cost  30  cents,  what  will  ^ 
cost  ?    What  will  ^  cost  ?     &  ?    &  ?    -&  ?    ^  ?    |§  '? 

21.  If  j.  of  an  acre  cost  8  28,  what  will  £  cost  ?     What 
will  an  acre  cost  ? 

22.  If  ^  of  a  share  in  a  railroad  be  worth  8  36,  what  is 
•f  worth  ?    What  is  the  value  of  a  whole  share  ? 

23.  When  ^T  of  a  share  in  a  factory  cost  8  60,  what  is 
the  value  of  T!T  ?     What  is  the  value  of  a  whole  share  ? 

24.  Gave  $  21  for  £  of  a  yard  of  broadcloth,  what  cost 
^  of  a  yard  ?    WThat  cost  a  yard  ? 

25.  Webster  paid  $8  for  f  of  a  chest  of  tea  ;  what  would 
^  of  a  chest  cost  ?     What  would  ^  of  a  £  cost  ?     What 
%  of  a  £  cost  ? 

26.  When  T*T  of  a  ton  of  iron  is  sold  for  8  32  ;  what  is 
the  cost  of  T1T  ?    Of  £  of  TV  ?    Of  %  of  TV  ? 

27.  Peter  Jones  paid   816  for  ^  of  an  ox  ;   what  cost 
y1^  of  the  ox,  and  what  did  Richard  Martin  pay  for  £  of 
a  y1^  ?  What  did  S.  Ayer  pay  for  a  ^  of  a  -^  ? 

28.  Paid  John  Atwood  $  128  for  |  of  his  farm  ;  what  is 
the  value  of  -i,  and  what  must  J.  Kimball  pay  for  £  of  a 
.1  ?    What  is  the  value  of  the  whole  farm  ? 

29.  I).  Webster  bought  ^  of  a  saw  mill,   for  which  he 
paid  8  300.     What  was  'the  value  of  the  whole  mill  ? 
What  is  the  value  of  f  of  the  mill  ?     Of  £  of  |  ?     Of  £ 
of  i  of  |  ? 

3D.   15  is  |  of  what  number  ?    Is  £  ?    Is  J  ?    Is  T3T  ? 


76  VULGAR   FRACTIONS.  [SECT.  20. 

31.  21  is  f  of  what  number  ?    Is  f  ?    Is  T%  ?    Is  ft  ? 

32.  30  is  T\  of  what  number  ?    Is  f  ?    Is  f  ?    Is  T\  ? 

33.  14  is  T2?  of  what  number  ?     Is  f  ?    Is  T2T  ?     Is  f  ? 

34.  12  is  T3g.  of  what  number  ?     Is  T3T  ?     Is  f  ?     Is  f  ? 

35.  18  is  T9T  of  what  number  ?     Is  -&  ?     Is  &  ?     Is  ,&  ? 

36.  Samuel  Page  sold  a  pair  of  oxen  for  $48,  which  was 
f-  of  their  cost.     What  did  he  lose  ? 

37.  Bought  a  horse   for  $  72,  which  was  f  of  his  real 
value  ;   what  did  I  gain  ? 

38.  72  is  f  of  what  number  ? 

39.  Sold  a  quantity  of  depreciated  money  for  $81,  which 
was  _?T  of  its  nominal  value  ;  what  was  the  sum  sold  ? 

40.  Having  improved  a  chaise  15  years,  it  was  sold  for 
$  25,  which  was  only  ^  of  what  it  cost.     What  was  the 
original  price  ? 

41.  A  Loafer  shot  at  a  flock  of  pigeons  on  a  tree,  and 
killed   24,   which   was  f  of  the   number.     How  many 
pigeons  will  remain  on  the  tree  ? 


Section  2O. 

VULGAR    FRACTIONS. 

FRACTIONS  are  parts  of  an  integer. 

VULGAR  FRACTIONS  are  expressed  by  two  terms,  called 
the  Numerator  and  Denominator  ;  the  former  above,  and 
the  latter  below  a  line. 

Thiia  •     $  Numerator     JL. 
InUS  ,     j  Denominator  1  !• 

The  Denominator  shows  into  how  many  parts  the  inte 
ger,  or  whole  number,  is  divided. 

The  Numerator  shows  how  many  of  those  parts  are 
taken. 

1.  A  proper  fraction  is  one  whose  numerator  is  less  than 

the  denominator  ,   as  f . 
£•  An  improper  fraction  is  one  whose  numerator  exceeds, 

or  is  equal  to,  the  denominator  ;   as  \^  or  f . 
3.  A  simple  fraction  has  a  numerator  and  denominator 

only  ;  as  f ,  y . 


SECT.  20.]  VULGAR   FRACTIONS.  77 

4.  A  compound  fraction  is  a  fraction  of  a  fraction,  con 
nected  by  the  word  of;  as  ^  of  |  of  §  of  £  . 

5.  A  mixed   number  is   an   integer  with  a  fraction  ;   as 

?T6T,  5i- 

6.  A  compound  mixed  fraction  is  one  whose  numerator  or 
denominator,  or  both,  is  a  mixed  number  ;  as  _!L   Or  —  . 

7.  The  greatest  common  measure  of  two  or  more  num 
bers  is  the  largest  number,  that  will  divide  them  without 
a  remainder. 

8.  The  least  common  multiple  of  two  or  more  numbers  is 
the  least  number,  that  may  be  divided  by  them  without 
a  remainder. 

9.  A  fraction  is  in  its  lowest  terms,  when  no  number  but 
a  unit  will  measure  both  its  terms. 

10.  A  prime  number  is  that  which  can  be  measured  only 
by  itself  or  a  unit  ;   as  7,  1  1  ,  and  19. 

11.  A  perfect  number  is  equal  to  the  sum  of  all  its  ali 
quot  parts  ;   as  6,  28,  496,  &c. 

12.  A  fraction  is  equal  to  the  number  of  times  the  nu 
merator  will  contain  the  denominator. 

13.  The  value  of  a  fraction  depends  on  the  proportion, 
which  the  numerator  bears  to  the  denominator. 

I.  To  find  the  greatest  common  measure  of  two  or  more 
numbers  ;  that  is,  to  find  the  greatest  number  that  will 
divide  two  or  more  numbers. 

1.  What  is  the  common  measure  of  84  and  132  ;  that  is, 
what  is  the  largest  number,  that  will  divide  both  of  these 
numbers  without  a  remainder  ?  Ans.  12. 


_ 

ident    it    will    also    divide   48, 

H4  which  is  equal  to  12  +  36.     It 

48)84(1  will   also  divide  84  ;    because 

48  84  is  equal  to  36  -j-  48  ;  for,  as 

36)48(1  12   will   divide    each   of  these 

3  6  numbers,    it   is   evident   it   will 

divide  their  sum.  For  the  same 
.,      .,,      ,         ,.   .,     ._.,. 
36       reason,  it  will  also  divide  132, 

which  is  equal  to  84  -(-48.  We 
therefore  find,  that  12  is  the  largest  number,  that  will  di- 

G* 


78  VULGAR    FRACTIONS.  [SECT.  20. 

vide  48  and  132  without  a  remainder.     It  is,  therefore,  its 
greatest  common  measure.     Hence  the  following 

RULE. 

Divide  the  greater  number  by  the  less,  and  if  there  be  a 
remainder,  divide  the  last  divisor  by  it,  and  so  continue  di 
viding  the  last  divisor  by  the  last  remainder,  until  nothing 
remains,  and  the  last  divisor  is  the  greatest  common  meas 
ure. 

If  there  be  more  than  two  numbers,  find  the  greatest  com 
mon  measure  of  two  of  them,  and  then  of  that  common 
measure  and  the  other  numbers.  If  it  should  happen,  that  1 
is  the  common  measure,  the  numbers  are  prime  to  each  other, 
and  are  incommensurable. 

2.  What  is  the  greatest  common  measure  of  85  and  95  ? 

Ans.  5. 

3.  What   is  the    greatest   common   measure  of  72  and 
168  ?  Ans.  24. 

4.  What   is  the  greatest   common  measure  of  119  and 
121  ?  Ans.  1. 

5.  What  is  the  largest  number  that  will  divide  324  and 
586  ?  Ans.  2. 

6.  What  is  the  largest  number  that  will  divide  582  and 
684  ?  Ans.  6. 

7.  What  is  the  greatest  common  measure  of  32  and  172  ? 

Ans.  4. 

8.  What  is  the  largest  number  that  will  divide  84  and 
1728  ?  Ans.  12. 

9.  What  is  the  greatest  common  measure  of  16,  20,  and 
26  ?  Ans.  2. 

10.  What  is  the  greatest  common  measure  of  12,  18,  24, 
and  30  ?  Ans.  6. 

II.  To  reduce  fractions  to  their  lowest  terms. 

NOTE.  A  fraction  is  said  to  be  in  its  lowest  terms,  when  no  num 
ber  but  a  unit  will  divide  its  numerator  and  denominator. 

1.  Reduce  T\  to  its  lowest  terms. 

OPERATION.  "We  find  by  the  last  Rule,  that  5  is 

5  )  ^  —  £  Ans.     the    largest   number,    that   will   divide 

both    the   numerator   and   denominator 

of  the  fraction  ;   and  having  divided  them  both  by  it,  we 


SECT.  20.]  VULGAR    FRACTIONS.  79 

find  the  result  to  be  ^-,  and  that  £  is  equal  to  T5^  is  evident 
from  the  fact,  that  the  ratio  of  5  to  15  is  equal  to  the  ra 
tio  of  1  to  3.  And,  as  the  value  of  a  fraction  depends  on 
the  ratio,  which  the  numerator  bears  to  the  denomina 
tor,  if  their  ratios  are  equal,  the  fractions  are  also  equal. 
Q.  e.  d.  Hence  the  following 

RULE. 

Divide  the  numerator  and  denominator  by  any  number 
that  will  divide  them  loth  without  a  remainder ;  and  so  con 
tinue  until  no  number  will  divide  them  but  unity.  Or,  di 
vide  the  numerator  and  denominator  by  the  greatest  common 
measure. 

2.  Reduce  ^  to  its  lowest  terms.  Ans.  £. 

3.  Reduce  -^  to  its  lowest  terms.  Ans.  f . 

4.  Reduce  ||  to  its  lowest  terms.  Ans.  \. 

5.  Reduce  T9¥6¥  to  its  lowest  terms.  Ans.  f . 

6.  Reduce  4-f  %  to  its  lowest  terms.  Ans.  £. 

7.  Reduce  -Jf|  to  its  lowest  terms.  Ans.  ^ff . 

8.  Reduce  -^V  to  its  lowest  terms.  Ans.  -$-. 

9.  Reduce  £ff&  to  its  lowest  terms.  Ans.  ^-ff&' 

10.  What  is  the  lowest  expression  of  fff  ?      Ans.  ^£f . 

III.  To  reduce  mixed  numbers  to  improper  fractions. 
MENTAL  OPERATIONS. 

1.  In  3  dollars  how  many  halves  ?    How  many  thirds  ? 

2.  In  7  apples  how  many  tenths  ?    How  many  twelfths  ? 

3.  In  8^-  dollars  how  many  sevenths  ? 

4.  In  3£  oranges  how  many  fourths  ? 

5.  In  9-j*r  gallons  how  many  elevenths  ? 

6.  In  7-|  quarts  how  many  fifths  of  quarts  ? 

OPERATION.          \ye  analyze  this  question  by  saying,  as 

there  are  5  fifths  in  one  quart,  there  will 

5  be  5  times  as  many  fifths  as  quarts  ;   there- 

~3~5  fore,  in  seven  quarts  and  three  fifths,  there 

3  will  be  38  fifths,  which  should  be  expressed 

^-5  thus,  M.     And  this  fraction,  by  definition 

2d,   on  page  76,   is  an  improper  fraction. 

5  Hence  the  following 


80  VULGAR    FRACTIONS.  [SECT.  20. 

RULE. 

Multiply  the  whole  number  by  the  denominator  of  tlie  frac 
tion,  and,  to  the  product  add  the  numerator,  and  place  their 
sum  over  the  denominator  of  the  fraction. 

7.  Reduce  8T3T  to  an  improper  fraction.  Ans.  ^i- 

8.  Reduce  15/2-  to  an  improper  fraction.         Ans.  ^/. 

9.  In  ]<•*£  how  many  ninths  ?  Ans.  -1!-. 

10.  In  IGl/jY  how  many  one  hundred  and  seventeenths  ? 

Ans.  i 

11.  Change  43f}^  to  an  improper  fraction    Ans. 

12.  What  improper  fraction  will  express  27T97  ? 

Ans. 

13.  Change  llly^y  to  an  improper  fraction  ? 

Ans. 


IV.  To  change  improper  fractions  to  integers  or  whole 
numbers. 

MENTAL  OPERATIONS. 

1.  How  many  dollars  in  4  halves  ?     In  5  halves  ?     In  6 
halves  ?    In  7  halves  ?    In  12  halves  ?    In  19  halves  ? 

2.  How  many  dollars  in  5  quarters  ?    In  9  quarters  ? 

3.  How  many  dollars  in  10  eighths  ?    In  20  eighths  ? 

FOR    THE    SLATE. 

4.  How  many  dollars  in  f  £  dollars  ?  Ans.  2T5F. 
OPERATION.               This  question  may  be  analyzed  by  say- 

1  6)  3  7  (2T5ff       ing,   as   16  sixteenths  make   one  dollar, 

3  2  there  will  be  as  many  dollars  in  37  six- 

5  teenths  as  37  contains  16,  which  is  2T5g- 

times,  =  $  2T5F.     This  answer  is  called 

a  mixed  number  by  definition  5th,   page  77.     Hence  we 

see  the  propriety  of  the  following 

RULE. 

Divide  the  numerator  by  the  denominator,  and  if  there 
be  a  remainder,  place  it  over  the  denominator  at  the  right 
hand  of  the  integer. 

5.  Change  l^  to  a  mixed  number.  Ans.  10T8T. 


SECT.  20.]  VULGAR    FRACTIONS.  81 


6.  Change  ^Vy1  to  a  mixed  number.  Ans. 

7.  Change  -W8^-  to  a  niixed  number.  Ans.  Ifyf. 

8.  Reduce  ^°-°  to  a  mixed  number.  Ans.  142f. 

9.  Reduce  |X|  to  a  whole  number.  Ans.  1. 

10.  Change  -^p-  to  a  whole  number.  Ans.  567. 

11.  What  is  the  value  of  -3^-  ?  Ans. 

12.  What  is  the  value  of  -£-5  ?  Ans. 

13.  Change  125  to  an  improper  fraction.         Ans.  J-f^. 

V.  To  change  or  reduce  compound  fractions  to  simple 
fractions. 

MENTAL  OPERATIONS. 

1.  What  part  of  an  orange  is  a  i  of  a  half  ? 

2.  What  part  of  an  apple  is  a  £  of  a  half  ? 

3.  What  part  of  a  bushel  is  a  ^  of  a  peck  ? 

4.  What  part  of  a  quart  is  a  ^  of  a  pint  ? 

FOR  THE  SLATE. 

5.  What  is  t  of  Vt  ?  Ans.  f  f  . 

OPERATION.  This  question  may  be  analyzed 

f  X  -fr  —  f  f  Ans.      by  saying,   if  -^  of  an   apple   be 

divided    into  5   equal   parts,   that 

one  of  these  parts  is  -^  of  an  apple  ;  and,  if  -£  of  ^  be 
•5^,  it  is  evident,  that  i  of  T7T  will  be  7  times  as  much.     7 
times  -5*5.  is  -^  ;  and,  if  ±  of  ^  be  ^,  f  of  ^  will  be  4 
times  as  much.     4  times  -fe  is  ff  . 
We  therefore  induce  the  following 

RULE. 

Change  mixed  nunibers  and  whole  numbers,  if  there  be 
any,  to  improper  fractions  ;  then  multiply  all  the  numera 
tors  together  for  a  new  numerator,  and  all  the  denominators 
together  for  a  new  denominator  ;  the  fraction  should  then 
be  reduced  to  its  lowest  terms. 

6.  What  is  §  of  $  of  f  ? 

OPERATION. 

f  X  f  X  f  =  T4^  =  if  Ans. 


82  VULGAR    FRACTIONS.  [SECT.  20. 

7.  What  is  }  of  T9T  of  7  ? 

OPERATION. 


8.  What  is  J  of  T9T  of  f  of  4.  ?  Ans.  ^  = 

9.  Change  ||  of  £  of  £  of  2V  of  7  to  a  simple  fraction. 

Ans. 


NOTE  1.     If  there  be  numbers  in  the  numerators  and  denominators, 
that  be  alike,  an  equal  number  of  the  same  value  may  be  cancelled. 

1O.  Reduce  £  of  f  of  -f  of  -/y  to  a  simple  fraction. 


STATEMENT.  CANCELLED. 


3X4X5X7        3X^X#X  tf         3, 

_  _  _  __  ___  _..  .      _  ATIQ 

4X5X7XH       4X0X*X  11       11 

In  performing  this  question,  we  perceive  that  there  is  a 
4  and  o  and  7  among  the  numerators,  and  also  the  same 
numbers  among  the  denominators  ;  these  we  cancel  before 
we  commence  the  operation. 

11.  Required  the  value  of  f  of  T4T  of  T|  of  J£  of  5J. 

STATEMENT. 

3x  4  x  11X17X23 


5x  11  X  17X23X  4 


CANCELLED. 


8  X  4  XteXtfXffi  _  3 

5  x  n  x  M  x  %fx~£  ~~  ~  5 

Reduce  -^  of  f  of  T9T  of  |  of  ^  to  a  simple  fraction. 


STATEMENT.  CANCELLED. 


1X8X  9  X5X3  _  3X^X0X^X3  _  _3_ 
5X^X11X8X7       0X0X11  X$X7~~77 

13.  Reduce  f  of  T*T  of  $  of  ^  of  4^  to  a  simple  fraction. 

Ans.  f|. 

NOTE  2.  When  there  are  any  two  numbers,  one  in  the  numerators 
and  the  other  in  the  denominators,  which  may  be  divided  by  a  num 
ber  without  a  remainder,  the  quotients  arising  from  such  division 
may  be  used  in  the  operation  of  the  question  instead  of  the  original 
numbers. 

14.  Reduce  |f  of  f  of  ^  to  a  simple  fraction. 


SECT.  20.]  VULGAR    FRACTIONS.  83 

STATEMENT.  CANCELLED. 

5 

15X8X7        If  X  9  X  7        35 

^^___^ ,      Ans 

16X9X11       10X9XH       66 
2       3 

In  performing  this  question,  we  find  that  the  15  among 
the  numerators  and  the  9  among  the  denominators  may  be 
divided  by  3,  and  that  the  quotients  will  be  5  and  3.  We 
write  the  5  above  the  15,  and  the  3  below  the  9.  We  also 
find  an  8  among  the  numerators,  and  a  16  among  the  de 
nominators,  which  may  be  divided  by  8,  and  that  the  quo 
tients  will  be  1  and  2.  We  write  the  1  over  the  8,  and 
the  2  under  the  16.  We  then  multiply  the  5,  and  1,  and 
7  together  for  a  new  numerator,  and  the  2,  and  3,  and  11 
together  for  a  new  denominator.  That  the  result  will  be 
the  same  by  this  process  as  by  the  other,  is  evident  from 
the  fact,  that  the  multiples  of  any  number  have  the  same 
ratio  to  each  other,  as  the  numbers  themselves. 

This  cancelling  principle,  when  well  understood,  will 
often  facilitate  the  operations  of  many  questions,  when 
the  divisors  and  dividends  have  a  common  denominator. 

15.  Reduce  T8T  of  f  f  of  ££  of  9|  to  a  whole  number. 

STATEMENT.  CANCELLED. 

3      tt 
8X22X^5X77        i" 


__  4 

~"     ~ 


11X35X22X8 


16.  Divide  the  continued  product  of  18,  24,  27,  and  30, 
by  the  continued  product  of  20,  21,  9,  and  10. 


STATEMENT.  CANCELLED. 


18X24X27X30      ;Tgxa*XgtXg0      324 

~        " 


20X21X  9  X10~#0x2lX  0  X*0~  35 
5711 

17.  Divide  the  continued  product  of  20,  19,  18,  17,  16, 
15,  14,  13,  12,  and  11,  by  the  continued  product  of  10, 
9,  8,  7,  6,  5,  4,  3,  2,  and  1. 


84  VULGAR    FRACTIONS.  [SECT.  20. 

CANCELLED. 

2  2  %         $       %  % 

20  X  19  X  &  X  17  X  *0  X  Tit  X  U  X  13  X  1$  X  11  _ 

*0x#x$xtfx0x£x4x#x2x*~ 

I         1         1         1         1         1  184756  Ans. 

NOTE.  In  this  question  the  product  of  the  quotients  of  2,  3,  2,  and 
2  is  cancelled  by  the  product  of  4,  3,  and  2  in  the  lower  line.  Any 
numbers  may  be  cancelled,  when  their  product  is  equal  to  the  product 
of  certain  other  numbers,  as  in  the  following  question. 

18.  Divide  the  continued  product  of  4,  9,  3,  8,  and  225 
by  the  continued  product  6,  6,  4,  6,  and  11. 

STATEMENT.  CANCELLED. 

4X9X3X8X225  _^X0X#X$X225_  225  _ 

6X6X4X6X  11  =  0X0X4X0X  H"  11  = 

20-jSj-  Ans. 

As  the  product  of  4  times  9  in  the  upper  line  is  equal 
to  the  product  of  6  times  6  in  the  under  line,  they  cancel 
each  other  ;  and  as  the  product  of  3  times  8  in  the  upper 
line  is  equal  to  4  times  6  in  the  under  line,  they  cancel 
each  other. 

VI.  To  find  the  least  common  multiple  of  two  or  more 
numbers,  that  is,  to  find  the  least  number,  that  may  be 
divided  by  them  without  a  remainder. 

RULE. 

Divide  by  such  a  number,  as  will  divide  most  of  the  giv 
en  numbers  without  a  remainder,  and  set  the  several  quo- 
tie?^  with  the  several  undivided  numbers  in  a  line  beneath, 
and  so  continue  to  divide,  until  no  number,  greater  than 
unity,  will  divide  two  or  more  of  them.  Then  multiply  all 
the  divisors,  quotients,  and  undivided  numbers  together,  and 
the  product  is  the  least  common  multiple. 

1.  What  is  the  least  common  multiple  of  8,  4,  3,  6  ? 
2)8436  ^  ls  evident,  that  24  is  a  corn- 

ox  T  ~~o — o — Q  posite  number,  and  that  it  is  com- 

^4     Z     6     6  posed  of  the  factors  2,   2,  3,  and 

^)  ^__J_?__2  ^  5    and»  therefore,  it  may  be   di- 

2111  vided  by  any  number,  which  is  the 

2X2X3X2  =  24  Ans. 


SECT.  20.]  VULGAR    FRACTIONS.  85 

product  of  any  two  of  them  ;  and,  as  the  given  numbers 
are  either  some  one  of  these,  or  such  a  number  as  may 
be  produced  by  the  product  of  two  or  more  of  them,  it  is 
evident,  therefore,  that  24  may  be  divided  by  either  of 
them  without  a  remainder.  Q.  e.  d. 

£.  What  is  the  least  common  multiple  of  7,  14,  21,  and 
15?  •  Ans.  210. 

3.  What  is  the  least  common  multiple  of  3,  4,  5,  G,  7, 
and  8  ?  Ans.  840. 

4.  What  is  the  least  number,  that  10,  12,  16,  20,  and  24 
will  divide  without  a  remainder  ?  Ans.  240. 

5.  Five  men  start  from  the  same    place  to  go  round  a 
certain  island.     The  first  can  go  round  it  in   10  days  ; 
the  second  in  12  days  ;   the  third  in  16  days  ;  the  fourth 
in    18  days  ;    the  fifth  in  20  days.      In  what  time  will 
they  all  meet  at  the  place  from  which  they  started  ? 

Ans.  720  days. 

VII.  To  reduce  fractions  to  a  common  denominator  ; 
that  is,  to  change  fractions  to  other  fractions,  all  having 
their  denominators  alike,  yet  retaining  the  same  value. 

1.  Reduce  f ,  £,  and  I  to  a  common  denominator. 
First  Method. 

OPERATION. 


4)468    4x2x3= 
2)162  4 


131 


24  common  denominator. 

~~6x3=18  numerator  for  f= 
4x5—20  numerator  for  f= 
3x7=21  numerator  for  = 


Having  first  obtained  a  common  multiple  of  all  the  de 
nominators  of  the  given  fractions  by  the  last  rule,  we  as 
sume  this,  as  the  common  denominator  required.  This 
number  (24)  we  divide  by  the  denominators  of  the  given 
fractions,  4,  6,  and  8,  and  find  their  quotients  to  be  6,  4, 
and  3,  which  we  place  under  the  24  ;  these  numbers  we 
multiply  by  the  numerators,  3,  5,  and  7,  and  find  their 
products  to  be  18,  20,  and  21,  and  these  numbers  are  the 
numerators  of  the  fractions  required. 


S6  VULGAR    FRACTIONS.  [SECT.  20. 

Second  Method. 


OPERATION. 


3  X  6  X  8  =  144  numerator  for  f  =  iff. 
5  x  4  X  8  =  160  numerator  for  £  =  |f  1 
7x4x6  =  168  numerator  for  %  —  |f  f . 

4  X  6  X  8  =  192  common  denominator. 

NOTE.  It  will  be  perceived,  that  this  method  does  not  express  the 
fractions  in  so  low  terras  as  the  other. 

From  the  above  illustration  we  deduce  the  following 
RULE. 

Let  compound  fractions  be  reduced  to  simple  fractions, 
mixed  numbers  to  improper  fractions,  and  whole  numbers  to 
improper  fractions,  by  writing  a  unit  under  them  ;  then 
Jlnd  the  least  common  multiple  of  all  the  denominators  by 
the  last  rule,  and  it  will  be  the  denominator  required.  Di 
vide  the  common  multiple  by  each  of  the  denominators,  and 
multiply  the  quotients  by  the  respective  numerators  of  the 
fractions,  and  their  products  will  be  the  numerators  required. 

Or,  multiply  each  numerator  into  all  the  denominators 
except  its  own  for  a  new  numerator ;  and  all  the  denomina 
tors  into  each  other  for  a  common  denominator. 

2.  Reduce  £  and  f  to  a  common  denominator. 

Ans.  fa  -f£. 

3.  Reduce  £,  T\,  and  ft.  Ans.  #«,  ^,  T\V 

4.  Reduce  f ,  T3¥,  and  2\.  Ans.  f  J,  fa  ££. 

5.  Reduce  fa  fa  and  £.  Ans.  if,  fa  if. 

6.  Change  &  fa  f ,  and  &.         Ans. 

7.  Change  f ,  |,  $,  and  £.  Ans. 

8.  Change  f ,  f ,  >,  and  -ft .      Ans.  |f  f 

9.  Reduce  f ,  ^»  and  7f  •  Ans.  J$,  f 

10.  Reduce  f ,  ^,  iJ,  and  5f          Ans.  if,  if,  ii,  -\/>/-. 

11.  Reduce  i,  f ,  f ,  f ,  },  and  ^.  Ans.  if,  if,  f  f ,  iJ,  J±,  if. 

12.  Change  |,  §,  £,£,£,  and  TV.  Ans.  if,  f|,^|,  ^,  &,,&. 

13.  Reduce  f ,  |,  and  TV  Ans.  ff ,  if,  JJ. 

14.  Change  7f ,  5T\,  7,  and  8.       Ans.  %i,  *-£ 

15.  Change  f ,  4,  5,  7,  and  9.        Ans.  f ,  J¥6-> 


SECT.  20.]  VULGAR    FRACTIONS.  87 

VITI.  To  reduce  fractions  of  a  lower  denomination  to  a 
higher. 

1.  Reduce  J  of  a  farthing  to  the  fraction  of  a  pound. 


OPERATION. 


£  X  f  qr.  r=  ^g-  =  |d.  This  question  may  be  an- 

^  v^  ^j   —  T^s.  alyzed   thus  ;    since   4   far- 

i    y     i    g  __    _i     ^  things  make  a  penny,  there 

will  be  •£•  as   many  pence  as 

farthings  ;  therefore  £  of  f-  of  a  farthing  is  •£•%  =  ^  of  a 
penny.  Again,  as  12  pence  make  a  shilling,  there  will 
be  y?  as  many  shillings  as  pence,  therefore  TV  of  ^  of  a 
penny  is  T^¥  of  a  shilling.  As  20  shillings  make  a  pound, 
there  will  be  5^  as  many  pounds  as  shillings,  therefore  ^ 
of  -j-^  of  a  shilling  is  -^-i^-^  of  a  pound.  Q.  e.  d. 

The  operation  of  this  question  may  be  abridged  thus  : 

OPERATION. 
4111  1 

_  VV      NX       >y/       ^^ 

f\     r1*         4     ^\       1    f~\      '^      /^f\    


9  12      20~2160 

Hence  the  following 

RULE. 

Let  the  given  fraction  le  reduced  to  a  compound  one  ly 
comparing  it  with  all  the  denominations  between  the,  given 
one  and  the  one  to  which  it  is  required  to  reduce  it ;  then 
reduce  this  compound  fraction  to  a  simple  one. 

2.  Reduce  $  of  a  grain  Troy  to  the  fraction  of  a  pound. 

4X1X1X1  1 

7X24X20X  12  ~~  100SO 

3.  What  part  of  an  ounce  is  T3a  of  a  scruple  ? 

0X1X1         1 

=  —  Ans. 


10X#X8       80 
4.  What  part  of  a  ton  is  ^  of  an  ounce  ? 

4X1X1X1X1  1 

5  X  16  X  28  X  4  X  20  ~~  44800 


88  VULGAR    FRACTIONS.  [SECT.  20. 

5.   What  part  of  a  mile  is  f  of  a  rod  ? 

0X1X1          I 

Ans. 


9  X  40  X  0       360 

6.  What  part  of  3  acres  is  ^  ">f  a  square  foot  ? 

9^<"2721~><40^2"X~3  =  294030  AnS' 

7.  What  part  of  3hhds.  is  ^  of  a  quart  ? 

4X1X1X1  1 

..  —      _ ,    '  •   .      A  r»a 

7X4X63X3       1323 

8.  What  part  of  3  yards  square,  are  3  square  yards  ? 

Ans   £. 

9.  What  part  of  £  of  a  solid  foot  is  £  of  a  foot  solid  ? 

Ans.  f. 

IX.  To  reduce  fractions  of  a  higher  denomination  to  a 
lower. 

1.  Reduce  TAir  of  a  pound  to  the  fraction  of  a  farthing. 

Ans.  §|. 

We  explain  this  question  in  the  following  manner. 

OPERATION.  As  shillings  are  twen- 

iiW  X  *r  --  yf  &IT  —  yVs-  tieths  of  a  pound,  there 

T\T  X  -1!2-  =  y§  —  -fs^-  will  be  20  times  as  many 

,s_  x  f  =  f  tqr.  Ans.  Parts   of    a    shilling    in 

T¥VTT    °^    a    pound,    as 

there  are  parts  of  a  pound  ;  therefore  y^Vir  °f  a  pound  is 
equal  to  y^tf  of  5T°-  =  yfw  —  yV  °^  a  shilling.  And  as 
pence  are  twelfths  of  shillings,  there  will  be  twelve  times 
as  many  parts  of  a  penny  in  T\j  of  a  shilling,  as  there  are 
parts  of  a  shilling  ;  therefore  T10  of  a  shilling  is  equal  to 
Td  °f  ~T~  —  T§  —  T&  °^  a  penny.  Again,  as  farthings  are 
fourths  of  a  penny,  there  will  be  4  times  as  many  parts 
of  a  farthing  in  76y  of  a  penny,  as  there  are  parts  of  a 
penny  ;  therefore  -fa  of  a  penny  are  equal  to  /F  of  £  =r: 
|i  of  a  farthing.  Q.  e.  d. 

The  operation  of  this  question  may  be  facilitated  by  the 
following  manner. 


OPERATION. 


X  -¥  X  Jf  X  t  =  T9A°o  = 


SECT.  20.]  VULGAR    FRACTIONS.  89 

Hence  the  following 

RULE. 

Let  the  given  numerator  be  multiplied  by  all  the  denom 
inations  between  it  and  the  one  to  which  it  is  to  be  reduced  ; 
then  place  the  product  over  this  denominator,  and  reduce  the 
fraction  to  its  lowest  terms. 

2.  What  part  of  a  grain  is  ^Vtf  of  a  pound  Troy  ? 

**Vo  X  V-  X  -2T°-  X  ¥  =  iif  £  =  f  Ans. 

3.  Reduce  T-g-Vtj  °f  a  furlong  to  the  fraction  of  a  foot. 

y?Vo  X  V-  X  ^  =  T6362°*  =  i  Ans. 

4.  What  part  of  a  square  foot  is  -j^Bir  °f  an  acre  ? 

innbn  X  *  X  -V-  X  *&  =  «»»  =  f 


5.  What  part  of  a  peck  is  -fy  of  a  bushel  ?          Ans.  y. 

6.  What  part  of  a  pound  is  j-fo-  of  a  cwt.  ?       Ans.  ^-|. 

X.  To  find  the  value  of  a  fraction  in  the  known  parts 
of  the  integer. 

RULE. 

Multiply  the  numerator  by  the  next  lower  denomination  of 
the  integer,  and  divide  the  product  by  the  denominator;  if 
any  thing  remains,  multiply  it  by  the  next  less  denomination, 
and  divide  as  before,  and  so  continue,  as  far  as  may  be  re 
quired  ;  and  the  several  quotients  will  be  the  answer. 

1.  What  is  the  value  of  /¥  of  a  pound  ?     Ans.  5s.  lOd. 

OPERATION. 
£.        8.  d. 

1     0       0 

7 


24)7     0       0 
0510 

What  is  the  value  of  £  af  a  cwt.  ? 

Ans.  3qr.  31b.  loz.  12fdr. 


90  VULGAR   FRACTIONS.  [SECT.  20. 

OPERATION. 
Cwt.   qr.    Ib.     oz.         dr. 

1000        0 

7 


9)7     0     0     0       0 
0     3    3     1     12| 

3.  What  is  the  value  of  £  of  a  yard  ?     Ans.  3qr.  Ofna. 

OPERATION. 
Yd.    qr.     na. 

100 

7 

9)7     0     0 
0     3     Of 

4.  What  is  the  value  of  ^  of  an  acre  ? 

Ans.  1R.  2Sp.  155ft.  82fin. 

OPERATION. 

A     R.        p.  ft.       in. 

100         00 
3 


7)3     00         00 
0     1   28    155 


5.  What  is  the  value  of  f  of  a  mile  ? 

Ans.  Ifur.  31rd.  1ft.  lOin. 

OPERATION. 

M.     fur.      rd.     ft.       in. 

10000 
2 


9)2     0000 
0     1    3  1     1    1  0 

6.  What  is  the  value  of  T3T  of  an  ell  English  ? 

Ans.  Iqr.  lT5Tna. 


SECT.  20.]  VULGAR    FRACTIONS.  91 

7.  What  is  the  value  of  f  of  a  hogshead  of  wine  ? 

Ans.  18gal.  Oqt.  Opt. 
S.  What  is  the  value  of  T7T  of  a  year  ? 

Ans.  232da.  lOh.  21m.  49-^sec. 

XI.  To  reduce  any  mixed  quantity  of  weights,  meas 
ures,  &,c.  to  the  fractions  of  the  integer.  » 

1.  What  part  of  a  pound  is  3s.  6d.  ? 

OPERATION.  To  perform  this   question, 

3s   6d  -  -    42d  we  reduce  the  3s.  (id.  to  pence, 

bein    the  lowest  denomina 


2Q  _  240d  ~ 

tion  in  the  question,   and  we 

make  them  the  numerator  of  the  fraction.  We  then  re 
duce  the  one  pound  to  pence,  and  make  them  the  denom 
inator  of  the  fraction.  This  fraction  we  reduce  to  its  low 
est  terms,  and  we  have  the  answer  required  ;  wherefore 
the  following 

RULE. 

Reduce  the  given  number  to  the  lowest  denomination  it 
contains  for  a  numerator,  and  reduce  the  integers  to  the 
same  denomination,  for  the  denominator  of  the  fraction 
required. 


Reduce  4s.  8d.  to  the  fraction  of  a  pound. 

N. 

- 


OPERATION. 

4s.  8d.  =    56d. 
20s.        =  2 

3.  What  part  of  a  ton  is  4cwt.  3qr.  121b.  ? 

OPERATION. 

4cwt.  3qr.  121b.  =    5441b.  _ 
20cwt.  =  224015;  ~~  ** 

4.  What  part  of  2m.  3fur.  20rd.  is  2fur.  30rd.  ? 

OPERATION. 

2fur.  30rd.  =  llOrd.  _,  ,   . 

2m.  3fur.  20rd.  =  TSOrd.  " 

5.  What  part  of  2A.  2R.  32p.  is  3R.  24  p.  ? 


92  VULGAR  FRACTIONS.  [SECT.  21 

OPERATION. 


. 

2A.  2R.  32P.  =  432P.  ""  * 

6.  What  part  of  a  hogshead  of  wine  is  18gal.  2qt.  ? 

Ans.  -&V 

7.  What  part  of  30  days  are  8  days  l?h.  20m.  ? 

Ans.  >  *£. 

8.  From   a  piece  of  cloth,   containing    13yd.   Oqr.   2na. 
there  were  taken  5yd.  2qr.  2na.     What  part  of  the  whole 
piece  was  taken  ?  Ans.  £. 


Section  21. 

ADDITION  OF  VULGAR  FRACTIONS. 

I.  To  add  fractions,  that  have  a  common  denominator. 

RULE. 

Write  the  sum  of  the  numerators  over  the  common  de 
nominator. 

1.  Add  },  %,  f ,  £,  and  f  together. 

OPERATION. 

1  +  2  -f-  4  +  5  +  6  =  V-  =  2f  Ans. 

2.  Add  T4T,  W,  -/T,  T8T,  T9r,  and  ft  together.     Ans.  3f }. 

3.  Add  T*T,  TsT,  T\,  TV,  and  ^  together.          Ans.  2Ty. 

4.  Add'^V,  r/V,  ^f,  and  f^  together.  Ans. 

5.  Add  ^,  |f,  ff,  and  f|  together.  Ans. 

6.  Add  ||f,  ^,  and  TyT  together.  Ans. 

7.  Add  iff f ,  tfft,  and  T|?T  together.        Ans.  1-^f |. 

II.  To  add  fractions  that  have  not  a  common  denomi 
nator. 

RULE. 

Reduce  mixed  numbers  to  improper  fractions,  and  com 
pound  fractions  to  simple  fractions ;   then  reduce  all  the 


SECT.  21.]  VULGAR    FRACTIONS.  93 

fractions  to  a  common  denominator  ;  and  the  sum  of  their 
numerators,  written  over  the  common  denominator^  will  be 
the  answer  required. 

1.  What  is  the  sum  of  f  ,  |,  and  ^  ? 


OPERATION. 


2)6    8    12    2X3X2X2  =  24  common  denominator 
3)3    4  ~6  6    4X5  =  20 

2)  1    4      2  81  3X3=    9 

~T~2 1  12    2X7=14 

4~3 

=  1££  Ans. 

2.  What  is  the  sum  of  |,  i£,  and  if  ?  Ans.  2£|. 

3.  Wrhat  is  the  sum  of  fut  ||,  and  •&  ?  Ans.  1  r52Yo-- 

4.  What  is  the  sum  of  £f ,  and  §|  ?  Ans.  If  If . 

5.  What  is  the  sum  of  f,  |,  f,  and  ^  ?  Ans.  2^. 

6.  Add  |,  58T,  ^,  and  £  together.  Ans.  lf§£. 

7.  Add  ||,  |^,  and  ^  together.  Ans.  l^jf£« 

8.  Add  ,&,  t§,  f  |,  and  TVo-  together.  Ans.  2f  § §. 

9.  Add  £,  f ,  f ,  |,  |,  f ,  and  £  together.  Ans. 

10.  Add  f,  r9jj,  ££,  |£,  if,  if,  and  |f  together. 

Ans. 

11.  Add  f  of  I  to  |  of  £.  Ans. 

12.  Add  }  of  I  to  i|  of  £.  Ans. 

13.  Add  ^  of  |  to  ^  of  T7o-.  Ans. 

14.  Add  f  of  f  of  |  to  |  of  f  of  •&.  Ans. 

15.  Add  ^  of  -ft  of  i£  to  £  of  f .  Ans. 

16.  Add  3^  to  4i£.  Ans. 

17.  Add  4|  to  5f .  Ans. 

18.  Add  17}  to  18-&.  Ans. 


NOTE  1.  If  the  quantities  are  mixed  numbers,  the  better  way  is  to 
add  the  fractional  parts  separately,  and  then  to  add  their  sum  to  the 
amount  of  the  whole  numbers. 

NOTE  2.  If  there  be  but  two  fractions  to  add,  and  their  numera 
tors  are  a  unit,  their  sum  may  be  found  by  writing  the  sum  of  the 


94  VULGAR   FRACTIONS.  [SECT.  22. 

denominators  over  their  product ;  thus, if  it  were  required  to  find  the 
sum  of  i  and  i,  we  should  add  the  3  and  7  together  for  a  numerator, 
and  multiply  them  together  for  a  denominator,  and  the  fraction  would 
be  10. 

19.  Add  i  to  i,  |  to  £,  *  to  £,  *  to  *,  J  to  i. 

20.  Add  £  to  A,  £  to  £,  |  to  J,  i  to  TV,  4  to  TV,  i  to  ^. 

21.  Add  |  to  |,  |  to  ^  ^  to  ^,  4  to  ^  4  to  4,  JT  to  A- 


Section  22. 

SUBTRACTION  OF  VULGAR  FRACTIONS. 

I.  To  subtract  fractions,  that  have  a  common  denom 
inator. 

RULE. 

Subtract  the  less  numerator  from  the  greater,  and  under 
the  remainder  write  the  common  denominator^  and  reduce 
the  fraction  if  necessary. 

OPERATION. 

1.  From  I  take  f.  7  —  2  =  5,  £  Ans. 

2.  From  T?r  take  T2T.  Ans.  ^. 

3.  From  -f  £  take  -fa.  Ans.  T4g-. 

4.  From  |f  take  fa.  Ans.  ff 

5.  From  jf  £  take  T^T.  Ans.  ||f-. 

6.  From  £fi*s  take  -tfrfr.  Ans.  f  gf . 

7.  From  ^  take  2^.  Ans.  -j^-. 

8.  From  TV^  take  y1^.  Ans.  f . 

II.  To  subtract  fractions  whose  denominators  are  unlike. 

RULE. 

Reduce  the  fractions  to  a  common  denominator,  as  in  Ad 
dition  of  fractions  ;  then  write  the  difference  of  the  nu 
merators  over  the  common  denominator. 


Sf.cr.  22.] 


VULGAR    FRACTIONS. 


95 


9.  From  j-f  take  -&.  Ans.  J|. 

OPERATION. 

4)  16     1  2     4x4x3=48  common  denominator. 


1O.  From  9f  take  5-J-J. 


16 
12 


3  x  13=  39 

4  X     7  =  28 

n 

48  Ans. 
Ans.  3ff 


OPERATION. 


4x2x3  = 

8 
12 


common  denominator. 


3X79  =  237 
2X71  =  142 


24 


11.  From  §  of  12£  take  f  of  9-&. 


=  3ff  Ans. 
Ans.  41. 


OPERATION. 


t  x  ¥  =  W,  f  x  -W  =  W  = 


8     1 

6x8x1  = 

4  8  common  denominator. 

48 

1  X  231=231 

6 

8  x     23=  1  84 

47 

4  8  Ans. 

12.  From  -/g-  take  /j-. 

Ans.  -5 

13.  From  £j  take  }%. 

Ans. 

14.  From  ££-  take  fa 

Ans.  - 

15.  From  •££  take  j^. 

Ans. 

16.  From  f  £  take  ^. 

Ans.  1 

17.  From  £f  take  T^-. 

Ans.  -t 

18.  From  £^£  take  T!g- 

Ans.  •§•; 

19.  From  ^  take  ^ 

j.                                    Ans.  yj 

96  VULGAR    FRACTIONS.  [SECT.  22. 

20.  From  f  of  T\  take  ±  of  f  Ans.  T7-&. 

21.  From  £  of  ^  take  *  of  if.  Ans.  TJT. 

22.  From  7£  take  3£.  Ans.  3f£. 

23.  From  8f  take  5|.  Ans.  2§f  . 

24.  From  9£  take  3£.  Ans.  5f  . 

25.  From  lOf  take  10^.  Ans.  f£. 

III.  To  subtract  a  proper  or  mixed  fraction  from  a 
whole  number. 

26.  From  16  take  1±.  Ans.  14£. 

OPERATION.  To  subtract  the  £  in  this  example,  1 

From       1  6  must   be  borrowed   from  the  6  in  the 

Take          l£         minuend,  and  reduced  to  fourths,  (J), 

"j^p"         and  the  £  must  be  taken  from  them  ;    £ 

from  j-  leaves  J.     To  pay  for  the    1, 

which  was  borrowed,  1  must  be  added  to  the  1  in  the  sub 

trahend,  1  -[-  1  =  2  ;    and  2  taken  from  16  leaves  14,  and 

the  f  ,  placed  at  the  right  hand  of  it,  gives  the  answer  14f  . 

The  same  result  will  be  obtained,  if  we  adopt  the  following 

RULE. 

Subtract  the  numerator  from  the  denominator  of  the  frac 
tion^  and  under  the  remainder  write  the  denominator,  and 
carry  one  to  the  subtrahend  to  be  subtracted  from  the  min- 
uend. 

OPERATION. 

27.          28.          29.  3O.  31. 

From      16  19  13  14  17 

Take         1  3 


f         15f  8«  6f         10* 

If  it  be  required  to  subtract  one  mixed  number  from 
another  mixed  number,  the  following  method  may  be 
adopted. 

32.  From  9f  take  3f  .  Ans.  5§|. 

OPERATION.  In    this    question,    we 

Minuend          9f  =  9^f  multiply  the  2  and  the  7, 

Subtrahend     3f  =  3§£  the    numerator    and    de- 

>24  A  nominator  of  the  fraction 

in  the  minuend  by  5,  the 


SECT.  22.]  VULGAR    FRACTIONS.  97 

denominator  of  the  fraction  in  the  subtrahend,  and  we 
have  a  new  fraction  ±%,  which  we  write  at  the  right  hand 
of  the  other  9,  thus,  9^-f .  We  then  multiply  the  numera 
tor  and  denominator  of  the  subtrahend  by  7,  the  denomi 
nator  of  the  minuend,  and  we  have  another  new  fraction, 
§|.  which  we  place  at  the  right  hand  of  the  other  3, 
thus,  3f£.  It  will  now  be  perceived,  that  we  have 
changed  the  fractions  9f  and  3f  to  other  fractions 
of  the  same  value,  having  a  common  denominator.  We 
now  subtract  as  in  question  26th  by  adding  1  (ff )  to  ££, 
which  makes  ||,  and  from  this  we  subtract  §£  ;  thus, 
^~| —  §i  rr  fi,  we  then  carry  the  1  we  borrowed  to  the  3, 
1  -(-  3  zrz  4,  which  we  take  from  9,  and  find  5  remaining. 
The  answer  therefore  is  ~~  w 

36.  37. 

18ft  19J* 
78H  6?f} 
41.  42. 

1  3/2-         1  5ft 

43.  From  a  hhd.  of  wine  there  leaked  out  12f  gallons, 
how  much  remained  ?  Ans.  50|. 

44.  From   $10,    $2|-   was  given  to   Benjamin,  $3£  to 
Lydia,  $  l£  to  Emily,   and  the  remainder  to  Betsey  ; 
what  did  she  receive  ?  Ans. 


NOTE.  If  it  be  required  to  find  the  difference  between  two  frac 
tions,  whose  numerators  are  a  unit,  the  most  ready  way  will  be  to 
write  the  difference  of  the  denominators  over  their  product. 

45.  What  is  the  difference  between  £  and  ^  ? 

OPERATION. 

7  —  3=     4 


46.  Take  |  from  £,  ^  from  £,  £  from  £,  ^  from  £. 

47.  Take  %  from  £,  £  from  \,  £  from  ^-,  -i  from  £. 

48.  Take  \  from  £,  \  from  4-,  TV  from  4-,  -Jy  from 


gg  VULGAR    FRACTIONS.  [SECT.  23. 

Section  33. 

MULTIPLICATION  OF  VULGAR  FRACTIONS. 

I.  To  multiply  a  fraction  by  a  whole  number,  or  a 
whole  number  by  a  fraction. 

Multiply  the  numerator  of  the  fraction  by  the  whole 
number,  and  under  the  product  write  the  denominator  of 
the  fraction. 

1.  Multiply  I  by  15. 

OPERATION.  This  question  may  be  an- 

j.  x  JL6.  —  J.<p.  —  J3j.  Ans.       alyzed   as    those    in   com 

pound  fractions. 

2.  Multiply  -H-  by  83. 

OPERATION. 


3.  If  a  man  receive  f  of  a  dollar  for  one  day's  labor, 
what  will  he  receive  for  21  days'  labor  ?        Ans.  $  ?£. 

4.  What  cost  561bs.  of  chalk  at  f  of  a  cent  per  Ib.  ? 

Ans.  $0.42. 

5.  What  cost  3961bs.  of  copperas  at  T9T  of  a  cent  per  Ib.  ? 

Ans.  $3.24. 

6.  What  cost  79  bushels  of  salt  at  £  of  a  dollar  per  bush 
el  ?  Ans.  $  G9£. 

7.  Multiply  376  by  ff  .  Ans.  243TST. 

8.  Multiply  if  by  189.  Ans.  166}f 

9.  Multiply  471  by  yfy.  Ans.    8F2g.. 

10.  Multiply  871  by  ^y.  Ans.  23ff 

11.  Multiply  fff  by  365.  Ans.  353TVT. 

12.  Multiply  867  by  T^.  Ans. 


II.  To  multiply  a  mixed  number  by  a  whole  number,  or 
a  whole  number  by  a  mixed  number. 

13.  Multiply  4f  by  7.  Ans.  32£. 


SECT.  23.]  VULGAR    FRACTIONS.  99 

OPERATION.  Jn  performing  this  question,  we  say  7 

4f  times  3  fifths  are  21   fifths,  and  21  fifths 

7  are  equal  to  4£.     We  write  down  the  -£ 

32.1  Ans.    and  carry  the  4  to  the  product  of  7  times 

4  =.  32.     Hence  the  following 

EULE. 

Multiply  the  numerator  of  the  mixed  number  by  the  whole 
number,  and  divide  the  product  by  the  denominator  of  the 
fraction  ;  and,  as  many  times  as  it  contains  the  denomina 
tor,  so  many  units  must  be  carried  to  the  product  of  the  in 
tegers.  If,  after  division,  any  thing  remains,  let  it  be  a 
numerator,  and  the  divisor  a  denominator  to  a  fraction  to 
be  affixed  to  the  product. 

14.  Multiply  9f  by  5.  Ans.  46£. 

15.  Multiply  12f  by  7.  Ans.  88f 

16.  Multiply  8|£  by  9.  Ans.  8()|. 

17.  Multiply  7£  by  10.  Ans.  71 J. 

18.  Multiply  llf  by  8.  Ans.  94f 

19.  What  cost  7T6Tlbs.  of  beef  at  5  cents  per  pound  ? 

Ans.  37T8T. 

20.  What  cost  23T72bbs.  flour  at  $  6  per  barrel  ? 

Ans.  8141J. 

21.  What  cost  8fyds.  cloth  at  $5  per  yard  ? 

Ans.  $41f. 

22.  What  cost  9  barrels  of  vinegar  at  $6f  per  barrel  ? 

Ans.  $5?f. 

23.  What  cost  12  cords  of  wood  at  $6.37£  per  cord  ? 

Ans.  $  76.50. 

24.  What  cost  llcwt.  of  sugar  at  8  9|  per  cwt.  ? 

Ans.  8  103^. 

25.  What  cost  4|  bushels  of  rye  at  $  1.75  per  bushel  ? 

Ans.  S7.(55f. 

26.  What  cost  7  tons  of  hay  at  $  11£  per  ton  ? 

Ans.  8  83|. 

27.  What  cost  9  doz.  of  adzes  at  $  lOf  per  doz.  ? 

Ans.  $95f. 

28.  What  cost  5  tons  of  lumber  at  $  3£  per  ton  ? 

Ans.  $  15£. 

29.  What  cost  locwt.  of  rice  at  $7.6^  per  cwt.  ? 

Ans.  $11 


100  VULGAR    FRACTIONS.  [SECT.  23. 

30.  What  cost  40  tons  of  coal  at  $  8.37£  per  ton  ? 

Ans.  8  335.00. 

III.  To  multiply  simple  fractions. 

31.  Multiply  $  by  f  .  Ans.  fe 

OPERATION.  This   question  may  be  anal- 

J  X  £  =  f  i  —  -/a  Ans.     yzed  in  the  same  manner  as  in 

compound  fractions. 
Hence  the  following 

RULE. 

Multiply  the  numerators  together  for  a  new  numerator, 
and  the  denominators  together  for  a  new  denominator  ;  then 
reduce  the  fraction  to  its  lowest  terms. 

32.  Multiply  J  by  TV  Ans.  T7T. 

OPERATION.  CANCELLED. 

78        56        7  7        $         7 


33.  Multiply  T5T  by  JJ.  Ans.  J. 

34.  Multiply  T%  by  £}.  Ans.  £. 

35.  Multiply  J|  by  **•  Ans.  f 

36.  Multiply  ty  by  ££.  Ans.  £. 

37.  Multiply  J  by  T8T.  Ans.  Tf  7. 

38.  Multiply  /T  by  f  f  .  Ans.  £. 

39.  What  cost  ^  of  a  bushel  of  corn  at  f  of  a  dollar  per 
bushel  ?  Ans.  J  of  a  dollar. 

40.  If  a  man  travels  T8T  of  a  mile  in  an  hour,  how  far 
would  he  travel  in  ^  of  an  hour  ?      Ans.  ^  of  a  mile. 

41.  If  a   bushel   of  corn  will  buy  y7^  of  a  bushel  of  salt, 
how  much  salt  might  be  bought  for  f  of  a  bushel  of 
corn  ?  Ans.  f^  of  a  bushel. 

NOTE.  If  there  be  mixed  numbers  in  the  question,  they  must  be 
reduced  to  improper  fractions,  and  compound  fractions  must  be  re 
duced  to  simple  fractions. 

42.  Multiply  4f  by  6|. 

OPERATION. 

4£  =  2£,  6|  =  %o_  ,  _¥  x  ^  =  W  =  30$  Ans. 


SECT.  23.]  VULGAR    FRACTIONS.  101 

43.  Multiply  7J  by  8f  Ans.  60^. 

44.  Multiply  4|  by  9£.  Ans.  45^. 

45.  Multiply  llf  by  8|.  Ans.  99-}l. 

46.  Multiply  12f  by  llf.  Ans.  14?£. 

47.  What  cost  7f  cords  of  wood  at  $  5f  per  cord  ? 

Ans.  $41f-J. 

48.  What  cost  7fyds.  of  cloth  at  $  3|  per  yard  ? 

Ans.  825|f. 

49.  What  cost  6f  gallons  molasses  at  23f-  cents  per  gal 
lon  ?  Ans.  8  15->Jf . 

50.  If  a  man  travels  3|  miles  in  one  hour,   how  far  will 
he  travel  in  9£  hours  ?  Ans.  34^. 

51.  What  cost  361^  acres  of  land  at  §25f  per  acre  ? 

Ans.  $91G7^i£. 

52.  If  f  of  £  of  a  dollar  buy  one  bushel  of  corn,  what 
will  £  of  T9T  of  a  bushel  cost  ?         Ans.  ^  of  a  dollar. 

53.  How  many  square  rods  of  land  in  a  garden,  which 
is  97T5F  rods  long,  and  49^  rods  wide  ? 

Ans.  4810-jkrods. 

54.  If  f  of  |-  of  -^j-  of  an  acre  of  land   cost   one  dollar, 
how  much  may  be  bought  with  §  of  $  18  ? 

Ans.  If^f-  acres. 

NOTE.   The  following  questions  are  to  exercise  the  foregoing  rules. 

55.  What  are  the  contents  of  a  field  76/^  rods  in  length 
and  IHf  rods  in  breadth  ?  Ans.  8A.  3R.  30|p. 

56.  What  are  the  contents  of  10  boxes  which  are  7 J  feet 
long,  If  wide,  and  1^  feet  in  height  ? 

Ans.  lC9i|-  cubic  feet. 

57.  From  T7T  of  an  acre  of  land  there  were  sold  20  poles 
and  200  square  feet.     What  quantity  remained  ? 

Ans.  2R.  lp.  22Jft. 

58.  What  cost  •££  of  an  acre  at  $  1.75  per  square  rod  ? 

Ans.  $  236.92TV 

59.  What  cost  T3F  of  a  ton  at  $  15f  per  cwt.  ? 

Ans.  $49.73^f. 

60.  What  is  the  continued  product  of  the  following  num 
bers  14f,  llf,  5|,  and  10£  ?  Ans.  9184. 

61.  From  T7j  of  a  cwt.  of  sugar  there  was  sold  f  of  it; 
what  is  the  value  of  the  remainder  at  $0.12-J  per  lb.? 

Ans.  $3.57. 
i* 


102  VULGAR  FRACTIONS.  [SECT.  24. 

62.  What  cost  19f  barrels  of  flour  at  $  7f  per  barrel  ? 

Ans.  $  143f . 

63.  What  cost  13T8T32-  quintals  offish  at  $3f  per  quintal? 

Ans.  flSlfJf 

64.  I  have  two  parcels  of  land,  one  containing  7T7^  acres, 
and  the  other  9^4-  acres.     What  is  their  value  at  $  78f 
per  acre  ?  Ans.  $  1380.70f . 

65.  From   a   quarter  of  beef  weighing  175£  Ibs.  I  gave 
John  Snow  f  of  it ;   f  of  the  remainder  I  sold  to  John 
Cloon.     What  is  the  value  of  the  remainder  at  8J  cents 
perlb.  ?  Ans.  $2.04^$. 

66.  Alexander  Green  bought  of  John  Fortune  a  box  of 
sugar  containing  475  Ibs.  for  $  30.00.     He  sold  £  of  it  at 
8  cents  per  lb.,  and  §  of  the  remainder  at  10  cents  per 
Ib.     What  is  the  value  of  what  still  remains  at  12£  cents 
per  lb.,  and  what  does  Green  make  on  his  bargain  ? 

A        (  Value  of  what  remains  $  13.19f. 
3<  }  Green's  bargain,  $  16.97f. 

67.  What  cost  y'/y  of  an  acre  at  $  14^  per  acre  ? 

Ans.  $2.00. 

68.  D.  Sanborn's  garden  is  23f  rods  long  and  13|  rods 
wide,  and  is  surrounded  by  a  good  fence  7|-  feet  high. 
Now  if  he  shall  make  a  walk  around  his  garden  within 
the  fence  7^  feet  wide,  how  much  will  remain  for  culti 
vation  ?  Ans.  1A.  3R.  7p.  85-j-^f^ft. 

69.  On  |  of  my  field,  I  plant  corn  ;   on  f  of  the  remain 
der  I  sow  wheat  ;   potatoes  are  planted  on  £  of  what  still 
remains,  and  I  have  left  two  small  pieces,  one  of  which 
is  3  rods  square,  and  the   other  contains  3  square  rods. 
How  large  is  my  field  ?  Ans    1A.  OR.  29p. 

70.  Multiply  |  of  T8T  of  H  by  T<y  of  JJ  of  Jf.     Ans.  TV 


Section  24. 

DIVISION  OF  VULGAR  FRACTIONS. 

I.  To  divide  a  fraction  by  a  whole  number. 
1.  How  many  times  will  ^  contain  9  ? 

OPERATION.  To    understand   this    question, 

we  will  suppose  f  of  an   apple 


SECT.  24.]  VULGAR   FRACTIONS.  103 

were  to  be  divided  equally  among  9  persons.  Now,  if 
we  divide  ^  of  an  apple  into  9  equal  parts,  there  would 
be  63  parts,  and  each  person  would  receive  fa  ;  but  there 
being  4,  each  man  will  receive  5  times  -^  =  557j  Ans. 
Hence  we  see  the  propriety  of  the  following 

RULE. 

Multiply  the  whole  number  by  the  denominator  of  the 
fraction,  and  write  the  product  under  the  numerator. 


2.  Divide  ^  by  12.  Ans. 

3.  Divide  ft  by  8.  Ans.  fa. 

4.  Divide  $  by  12.  Ans.  T£¥. 

5.  John  Jones  owns  ^  of  a  share  in  a  railroad  valued  at 
$117;   this  he  bequeaths  to  his  five  children.     What 
part  of  a  share  will  each  receive  ?  Ans.  ^. 

6.  Divide  &  by  15.  Ans.  Tf^. 

7.  Divide  T6T  by  28.  Ans.  ?f  ¥. 

8.  James  Page's  estate  is  valued  at  810,000,  and  he  has 
given  f  of  it  to  the  Seamen's  Society  ;  ^  of  the  remain 
der.  he  gave  to  his  good  minister  ;   and  the  remainder  he 
divided    equally    among    his   4    sons    and    3   daughters. 
What  sum  will  each  of  his  children  receive  ? 

Ans.  $680T\°T. 

II.  To  divide  a  whole  number  by  a  fraction. 

9.  How  many  times  will  13  contain  f-  ?  Ans.  30£. 

OPERATION.  It   is    evident,    that    13 

-lj-3-  X  I  —  Q  =  30¥  Ans-          wil1   contain   -f  ,    as  many 

times  as  there  are  sevenths 

in  13,  which  are  7  X  13  =  91  times.  Again,  if  13  con 
tain  1  seventh  91  times,  it  will  contain  3  sevenths  as 
many  times  as  91  will  contain  3  =  30^  Ans.  Hence  the 
following 

RULE. 

Multiply  the  whole  number  by  the  denominator  of  the 
fraction,  and  divide  the  product  by  the  numerator. 

10.  Divide  18  by  J.  Ans. 

11.  Divide  27  by  -&.  Ans. 


104  VULGAR    FRACTIONS.  [SECT.  24. 

12.  Divide  23  by  J.  Ans.  92. 

13.  Divide  5  by  £.  Ans.  25. 

14.  Divide  12  by  f  .  Ans.  16. 

15.  Divide  16  by  £.  Ans.  32. 

16.  Divide  100  by  |J.  Ans.  lll|f 

17.  I  have  50  square  yards  of  cloth,  how  many  yards,  f 
of  a  yard  wide,  will  be  sufficient  to  line  it  ? 

Ans.  83£  yards. 

18.  A.  Poor  can  walk  3T7T  miles  in  60  minutes  ;   Benja 
min  can  walk  T9T  as  fast  as  Poor.     How  long  will  it  take 
Benjamin  to  walk  the  same  distance  ? 

Ans.  73^  minutes. 

III.  To  divide  a  mixed  number  by  an  integer. 

19.  Divide  17£  by  6.  Ans.  2f  f  . 

OPERATION.  \ye  divide  17  by  6,  and  find  it  is  con- 

6)  1  7f  tained  2  times,  which  we  write  under  the 

24  3  17,  and  we  have  5  remaining,  which  we 

multiply    by   8,  the   denominator  of  the 

fraction  ;   and  to  the   product  we   add  the  numerator,  3, 

and  the  amount  is  43,  this  we  write  over  the  product  of  6, 

the  divisor,  multiplied  by  the  denominator,  8,  =  48.     The 

rationale  of  the  above  question  is  the  same  as  of  those  in 

Rule  I.  of  this  section.     Hence  the  following 

RULE. 

Divide  the  integers  as  in  whole  numbers,  and  if  any  thing 
remains,  multiply  it  by  the  denominator  of  the  fraction, 
and  to  the  product  add  the  numerator  of  the  fraction,  and 
write  it  over  the  product  of  the  divisor,  multiplied  by  the 
denominator. 


20.  Divide  17f  by  7.  Ans. 

21.  Divide  18£  by  8.  Ans. 

22.  Divide  27}£  by  9.  Ans.  3TVF. 

23.  Divide  31TV  by  11.  Ans.  2T9TV 

24.  Divide  78f  by  12.  Ans. 

25.  Divide  189||  by  4.  Ans. 

26.  Divide  107TV  by  3.  Ans.  35|f  . 


SECT.  24.]  VULGAR    FRACTIONS.  105 


27.  Divide  $  175  among  7  men.  Ans. 

28.  Divide  $  106£  among  8  boys.  Ans.  8  13f|. 

29.  What  is  the  value  of  f  f  of  a  dollar  ? 

Ans.  8  0.34  jf. 

SO.  Divide  $  107T7T  among  4  boys  and  3  girls,  and  give 
the  girls  twice  as  much  as  the  boys. 

Ans.  boy's  share  $  lOff.     Girl's  share  $21ff. 

31.  If  $  14  will  purchase  JJ  of  a  ton  of  copperas,  what 
quantity  will  8  1  purchase  ?         Ans.  Icwt.  Oqr.  241bs. 

IV.  To  divide  one  fraction  by  another. 

32.  Divide  J-  by  f,  Ans.  l£i. 

OPERATION.  To    understand   the    ra- 

J  X  f  =  ff  —  If?  Ans.  tionale  of  this  process,  we 

find  the  two  factors   of   f-, 

which  are  f  and  J  ;  for  £  multiplied  by  ^  are  f-,  as  is 
evident  from  a  preceding  rule.  We  now  divide  |-  by  \, 
which,  by  case  I.  of  this  section,  will  be  |-  X  T  =  irV 
Again,  we  wish  to  divide  -j-%  by  ^.  It  is  evident,  that  ^ 
will  contain  ^  nine  times  as  often,  as  it  will  a  unit,  and  ft 
contains  a  unit  -j^  times,  therefore  it  contains  ^  nine  times 
772-  =  f  X  -fa  =  f  f  =  1$J  Ans.  In  performing  this  ques 
tion,  it  will  be  perceived,  that  the  numerator  of  the  divi 
dend  has  been  multiplied  by  the  denominator  of  the  di 
visor,  and  the  denominator  of  the  dividend  by  the  numera 
tor  of  the  divisor.  Hence  the  following 

RULE. 

Invert  the  divisor  and  proceed  as  in  multiplication.  If, 
however,  there  be  mixed  numbers  in  the  question,  they  must 
be  reduced  to  improper  fractions,  and  compound  fractions 
must  be  reduced  to  simple  fractions. 

33.  Divide  J  by  f 

OPERATION. 


34.  Divide  7J  by  3f  . 

OPERATION. 

7f  =  -V-,  3f  =  ^,  JTL  X  ^  =  ttf.  =  2f  f  Ans. 

35.  Divide  J  by  J.  Ans.  3J-. 


106  VULGAR  FRACTIONS.  [SECT.  25. 

36.  Divide  f f  by  j-J.  Ans.  \\. 

37.  Divide  f  by  TV  Ans.  2f. 

38.  Divide  -^  by  f .  Ans.  6-ft. 

39.  Divide  f  by  T2T.  Ans.  4f . 

40.  Divide  7f  by  4j,  Ans.  Iff. 

41.  Divide  3£  by  7f  Ans.  TV 

42.  Divide  ll£  by  5f  Ans.  2T^. 

43.  Divide  4f  by  1£.  Ans. 

44.  Divide  11 6f  by  14f  Ans. 

45.  Divide  Slf  by  9f  Ans.  8|§2. 

46.  Divide  f  of  £  by  \  of  f.  Ans.  18f . 


Section 

EXERCISES  IN  VULGAR  FRACTIONS. 

I.  What  are  the  contents  of  a  board  9  inches  long  and 
7  inches  wide  ?  Ans.  63  square  inches. 

£.   What  are  the  contents  of  a  board  llf  inches  long,  and 
4£  inches  wide  ?  Ans.  49J-£  square  inches. 

3.  How   many   square  rods   in   a  garden,    which    is   iSf 
rods  in  length  and  9^  rods  wide  ?      Ans.  178^$  rods. 

4.  What  cost  19f  acres  of  land,  at  $  17f  per  acre  ? 

Ans.  $350^. 

5.  What  cost  14?7a  tons  of  coal  at  $  7f  per  ton  ? 

Ans.  8111JJ. 

6.  What  cost  13££  tons  of  hay  at  $  8J  per  ton  ? 

Ans.  $  120T\V 

7.  What  cost  1 J  bushels  of  corn  at  $  1J-  per  bushel  ? 

Ans.  $  3|f . 

8.  What  is  the  value  of  ^  of  a  dollar  ?     Ans.  $  0.56£. 

9.  What  is  the  value  of  |&  of  a  dollar  ?     Ans.  $0.2l£. 

10.  What  is  the  value  of  TV«y  of  a  dollar  ?   Ans.  $0.25f. 

II.  What  is  the  value  of  j{£  of  a  dollar  ?  Ans.  $05IT9F. 
12.  Bought  a  cask  of  molasses,  containing  871  gallons  ; 

f  of  it  having  leaked  out,  the  remainder  was  sold  at  2 
cents  per  gallon  ;  what  was  the  sum  received  ? 

Ans.  $  io.03p. 


SECT.  25.]  VULGAR    FRACTIONS.  107 

13.  Bought  of  L.  Johnson  Tfyds.  of  broadcloth,  at  $3| 
per  yard,  and  sold  it  at  §4§-  per  yard  ;  what  was  gained  ? 

Ans.  $3.08£. 

14.  Bought  a  piece  of  land,  that  was  47T5T  rods  in  length, 
and  29T7F  in  breadth  ;    and   from  this  land,  there  was 
sold  to  Abijah   Atwood  5  square   rods,  and  to  Hazen 
Webster  a  piece  that  was  5  rods    square  ;    how  much 
remains  unsold  ?  Ans.  1360||  square  rods. 

15.  Bought  a  tract  of  land  that  was  97  rods  long  and 
48T£  rods  wide  ;   and  from  this  I  sold  to  John  Ayer,  a 
houselot,  18T5rr  rods   long,  and    14^  rods  wide  ;   and  the 
remainder  of  my  purchase  was  sold  to  John  Morse,  at 
$3.75  per  square  rod  ;  what  sum  shall  I  receive  ? 

Ans.  $16717.30£f. 

16.  What  are  the  contents  of  a  box  8  feet  long,  5  feet 
wide,  and  3  feet  high  ?  Ans.  120  solid  feet. 

17.  What  are  the  contents  of  10  boxes,  each  of  which  is 
7|-  feet  long,  4^  feet  wide,  and  3|  feet  high  ? 

Ans.  1312^  feet. 

18.  Polly  Brown  has    $  17.S7£  ;    half   of  this  sum  was 
given  to  the  missionary  society,  and  f  of  the   remainder 
she  gave  to  the  Bible  society  ;  what  sum  has  she  left  ? 

Ans.  $3.57£. 

19.  What  number  shall  be  taken  from  12|,  and  the  re 
mainder  multiplied  by  10£  that  the  product  shall  be  50  ? 

Ans.  8TW- 

20.  What  number  must   be   multiplied    by  7|,  that  the 
product  may  be  20  ?  Ans  2if . 

21.  Bought  of  John  Dow  9£  yards  of  cloth  at   8  4.62£ 
per  yard  ;   what  was  the  whole  cost  ?     Ans.  §  45.67r3F. 

22.  Bought  of  John  Appleton  47|-  gallons  of   molasses 
for    8  12.37^  ;    what  cost   one  gallon  ?    what   cost   12£ 
gallons  ?  Ans   $  3.33f  £f . 

23.  When   $  15.87J   are  paid  for  12f  bushels  of  wheat, 
what  cost  one  bushel  ?   what  cost  11  bushels  ? 

Ans.  $  14.11£. 

24.  When    $  19.18f    are    paid    for   3f  cords   of  wood, 
what  cost  one  cord  ?   what  cost  f  of  a  cord  ? 

Answer  to  the  last,  $2  13^. 

25.  What  are  the  contents  of  a  box  8-&  feet  long, 
feet  wide,  and  2-^  feet  high  ?  Ans.  68T|^  feet. 


108  DECIMAL    FRACTIONS.  [SECT.  26. 

Section  £6. 

DECIMAL    FRACTIONS. 

A  DECIMAL  FRACTION  is  that,  whose  integer  is  always 
divided  into  10,  100,  1000,  &c.  equal  parts.  Its  denomi 
nator  is  always  an  unit,  with  as  many  ciphers  annexed, 
as  there  are  places  in  the  given  decimal.  There  is,  there 
fore,  no  need  of  having  the  denominator  expressed  ;  for 
the  value  of  the  fraction  is  always  known  by  placing  a 
point  before  it,  at  the  left  hand,  called  the  separatrix. 
Thus,  .5  is  T^,  .37  is  -j^V,  .348  is  fiffo. 

Ciphers  annexed  to  the  right  hand  of  decimals  do  not 
increase  their  value  ;  for  .4  or  .40  or  .400  are  decimals 
having  the  same  value,  each  being  equal  to  -^5-  or  f  ;  but 
when  ciphers  are  placed  on  the  left  hand  of  a  decimal, 
they  decrease  the  value  in  a  tenfold  proportion.  Thus 
.4  is  -j^,  or  four  tenths  ;  but  .04  is  T^,  or  four  hun- 
dredths  ;  and  .004  is  y^^-,  or  four  thousandths.  The 
figure  next  the  separatrix  is  reckoned  so  many  tenths  ; 
the  next  at  the  right,  so  many  hundredths;  the  third  is  so 
many  thousandths  ;  and  so  on,  as  may  be  seen  by  the  fol 
lowing 

TABLE. 

4 


H       a  ^5        3 

,  :  I  n      1  1  1  1  1 

BQ  Q         (**  5  Q  •O>crtF£O>±J 

S3          £         O          $         £  GO         ^          M        P-H          **         S 

ill  JlllilJIil 

7654321.      234567 

From  this  table  it  is  evident,  that  in  decimals,  as  well 
as  in  whole  numbers,  each  figure  takes  its  value  by  its 
distance  from  the  place  of  units. 


SECT.  27.]  DECIMAL    FRACTIONS.  JQQ 

NOTE.  If  there  be  one  figure  in  the  decimal,  it  is  so  many  tenths; 
if  there  be  two  figures,  they  express  so  many  hundredths  j  if  there  be 
three  figures,  they  are  so  many  thousandths,  &c. 

NUMERATION  OF  DECIMAL   FRACTIONS. 

Let  the  pupil  write  the  following  numbers. 

1.  Three  hundred  seven,  twenty-five  hundredths. 

2.  Forty-seven,  and  seven  tenths. 

3.  Eighteen  and  five  hundredths. 

4.  Twenty-nine  and  three  thousandths. 

5.  Forty-nine  ten  thousandths. 

6.  Eight  and  eight  millionths. 

7.  Seventy-five  and  nine  tenths. 

8.  Two  thousand  and  two  thousandths. 

9.  Eighteen  and  eighteen  thousandths. 

10.  Five  hundred  five,  and  one  thousand  six  millionths. 


Section  27. 

ADDITION    OF    DECIMALS. 

1.    Add    together    5.018;    171,16;     88.133;     1113.6; 
.00456,  and  14.178. 

OPERATION. 

5.0  1  8  =  Five  and  eighteen  thousandths. 

171.16  =  One  hundred  seventy-one,  sixteen  hundredths. 

88.133  =.  Eighty-eight,  and  one  hundred  thirty-three  thousandths, 

111  3.6  =z  One  thousand  one  hundred  thirteen,  and  six  tenths. 

.00456  —  Four  hundred  fifty-six  hundred  thousandths. 

1  4.1  7  O  =  Fourteen,  and  one  hundred  seventy-eight  thousandths. 

1QQOnOQ£ft    One   thousand    three   hundred    ninety-two,    and    nine 
oy^.uyrfOO   -         thousand  three  hundred  fifty-six  hundred  thousandths. 

RULE. 

Write  the  numbers  under  each  other  according  to  their 
value,  add  as  in  whole  numbers,  and  point  off  from  the  right 
hand  as  many  places  for  decimals,  as  there  are   in  that 
number,  which  contains  the  greatest  number  of  decimals, 
j 


HO  DECIMAL    FRACTIONS.  [SECT.  28. 

2.  Add  together  171.61111  ;   16.7101  ;  .00007  ;  71.0006, 
and  1.167895.  Ans.  260.489775. 

3.  Add    together    .16711;     1.766;     76111.1;     167.1; 
.000007,  and  1476.1.  Ans.  77756.233117. 

4.  Add  together  151.01  ;  611111.01  ;   16.5  ;  6.7  ;  46.1, 
and  .67896.  Ans.  611331.99896. 

5.  Add  fifty-six  thousand  and  fourteen  thousandths,  nine 
teen  and  nineteen  hundredths,  fifty-seven  and  forty-eight 
ten  thousandths,  twenty-three  thousand  and  five  and  four 
tenths,  and  fourteen  millionths.        Ans.  79081.608814. 

6.  What  is  the  sum  of  forty-nine  and  one  hundred  and 
five  ten  thousandths,  eighty-nine  and  one  hundred  seven 
thousandths,  one  hundred  and  twenty-seven  millionths, 
forty-eight  ten  thousandths  ?  Ans.  138.122427. 

*7.  What  is  the  sum  of  three  and  eighteen  ten  thousandths, 
one  thousand  five  and  twenty-three  thousand  forty-three 
millionths,  eighty-seven  and  one  hundred  seven  thou 
sandths,  forty-nine  ten  thousandths,  and  forty-seven  thou 
sand  and  three  hundred  nine  hundred  thousandths  ? 

Ans.  48095.139833. 


Section  28. 

SUBTRACTION    OF    DECIMALS. 

RULE. 

Let  the  numbers  be  so  written  that  the  separatrix  of  the 
subtrahend  be  directly  under  that  of  the  minuend,  that  is, 
units  under  units,  and  tens  under  tens,  fyc. ;  subtract  as  in 
whole  numbers,  and  point  off  so  many  places  for  decimals, 
as  there  are  in  that  number,  which  contains  the  greatest  num 
ber  of  decimals. 

OPERATION. 

2.  3.  4. 

47.117  46.13  87.107 

8.78195  7.8915  1.1  1986 

1.268          38.33505          382385         85.98714 


SECT.  29.]  DECIMAL    FRACTIONS.  Ill 

5.  From  81.35  take  11.678956.  Ans.  69.671044. 

6.  From  1.  take  .876543.  Ans.  .123457. 

7.  From  100.  take  99.111176.  Ans.  .8888-24. 

8.  From  87.1  take  5.6789.  Ans.  81.4211. 

9.  From  100.  take  .001.  Ans.  99.999. 

10.  From  seventy-three  take  seventy-three  thousandths. 

Ans.  72.927. 

11.  From  three  hundred  sixty-five  take  forty-seven  ten 
thousandths.  Ans.  364.9953. 

12.  From  three  hundred  fifty-seven  thousand  take  twenty- 
eight  and  four  thousand  nine  ten  millionths. 

Ans.  356971.9995991. 

13.  From  .875  take  .4.  Ans.  .475. 

14.  From  .3125  take  .125.  Ans.  .1875. 

15.  From  .95  take  .44.  Ans.  .51. 

16.  From  3.7  take  1.8.  Ans.  1.9. 

17.  From  8.125  take  2.6875.  Ans.  5.4:*75. 

18.  From  9.375  take  1.5.  Ans.  7.875. 

19.  From  .666  take  .041.  Ans.  .625. 


Section  29. 

MULTIPLICATION  OF   DECIMALS. 
1.  Multiply  18.72  by  7.1.  Ans.  132.912. 

OPERATION  BY  DECIMALS.       BY  YULGAR  FRACTIONS. 

18.72 


Ans.  X  »  = 

Multiply  15.12  by  .012.  Ans.  .18144. 

OPERATION    BY  DECIMALS.  BY   TULGAR    FRACTIONS. 

15.12 


_ 

3~0  24  -012  =  T**IF 

.rll44  Ans.  -W  X  i  ifo  =  yVV^  Ans 


112  DECIMAL   FRACTIONS.  [SECT.  29. 

Hence,  we  deduce  the  following 
RULE. 

Multiply  as  in  whole  numbers,  and  point  off  as  many 
figures  for  decimals  in  the  product,  as  there  are  decimals 
in  the  multiplicand  and  multiplier ;  but,  if  there  be  not  so 
many  figures  in  the  product,  as  in  the  multiplicand  and 
multiplier,  supply  the  defect  by  prefixing  ciphers. 

3.  Multiply  18.07  by  .007.  Ans.. 12649. 

4.  Multiply  18.46  by  1.007.  Ans.  18.58922. 

5.  Multiply  .00076  by  .0015.  Ans.  .00000114. 

6.  Multiply  11.37  by  100.  Ans.  1137. 

7.  Multiply  47.01  by  .047.  Ans.  2.20947. 

8.  Multiply  .0701  by.  0067.  Ans.  .00046967. 

9.  Multiply  47.  by  .47.  Ans.  22.09. 

10.  Multiply    eighty-seven   thousandths    by    fifteen   mil- 
lionths.    J  Ans.  .000001305. 

11.  Multiply  one    hundred   seven   thousand    and    fifteen 
ten  thousandths  by  one  hundred  seven  ten  thousandths. 

Ans.  1144.90001605. 

12.  Multiply  ninety-seven  ten  thousandths  by   four  hun 
dred  and  sixty-seven  hundredths.  Ans.  3.886499. 

13.  Multiply  ninety-six  thousandths    by  ninety-six    hun 
dred  thousandths.  Ans.  .00009216. 

14.  Multiply  one  million  by  one  millionth.  Ans.  1. 

15.  Multiply  one  hundred  by  fourteen  ten  thousandths. 

Ans.  .14. 

16.  Multiply  one  hundred  and  one  thousandth     by  ten 
thousand  one  hundred  one  hundred  thousandths. 

Ans.    .01020201. 

17.  Multiply  one  thousand  fifty  and  seven  ten  thousandths, 
by  three  hundred  five  hundred  thousandths. 

Ans.  3.202502135. 

18.  Multiply  two  million  by  seven  tenths. 

Ans.  1400000. 

19.  Multiply  four  hundred  and  four  thousandths  by  thirty 
and  three  hundredths.  Ans.  12012.12012. 

20.  What  cost  461bs  tea  at  $  1.125  per  Ib.  ?      $51.75. 

21.  What  cost  17.125  tons  of  hay  at   $  18  875  per  ton  ? 

Ans.  $  323.234375. 

22.  What  cost  181bs.  sugar  at  $  .125  per  Ib.  ? 

Ans.  $2.25. 


SECT.  30.]  DECIMAL    FRACTIONS.  113 

Section  3O. 

DIVISION    OF    DECIMALS. 
1.  Divide  $45.625  by  12.5.      2.  Divide  45T6^  by 

OPERATION   BY   DECIMALS.  BY   VULGAR   FRACTIONS. 

1  2.5)  4  5.6  2  5  (3.6  5 
375 


625 

Hence  the  following 

RULE. 

Divide  as  in  whole  numbers,  and  point  off"  as  many  deci 
mals  in  the  quotient,  as  the  number  of  decimals  in  the  divi 
dend  exceed  those  of  the  divisor  ;  but,  if  the  number  of  those 
in  the  divisor  exceed  that  of  the  dividend,  reduce  the  divi 
dend  to  the  same  denomination  as  the  divisor  by  annexing 
ciphers.  And,  if  the  number  of  decimals  in  the  quotient 
and  divisor  together  are  not  equal  to  the  number  in  the  divi 
dend,  supply  the  defect  by  prefixing  ciphers  to  the  quotient. 

3.  Divide  183.375  by  489.  Ans.  .375. 

4.  Divide  67.8632  by  32.8.  Ans.  2.069. 

5.  Divide  67.56785  by  .035.  Ans.  1930.51. 

6.  Divide  .567891  by  8.2.  Ans.  .069255. 

7.  Divide  .1728  by  12.  Ans.  .0144. 

8.  Divide  172.8  by  1.2.  Ans. 

9.  Divide  1728.  by  .12.  Ans. 

10.  Divide  .1728  by  .12.  Ans. 

11.  Divide  1.728  by  12.  Ans. 

12.  Divide  17.28  by  1.2.  Ans. 

13.  Divide  1728  by  .0012.  Ans. 

14.  Divide  .001728  by  12.  Ans. 

15.  Divide  one  hundred  forty-seven  and  eight  hundred 
twenty-eight  thousandths  by  nine  and  seven  tenths. 

Ans.  15.24. 

T    - 


114  DECIMAL    FRACTIONS.  [SECT.  31. 

16.  Divide  six  hundred  seventy-eight  thousand  seven 
hundred  sixty-seven  millionths  by  three  hundred  twenty- 
eight  thousandths.  Ans.  2.069. 


Section  31* 

REDUCTION   OF   DECIMALS. 

I.  To  reduce  a  vulgar  fraction  to  a  decimal. 

1.  Reduce  f  to  a  decimal. 

OPERATION.  That  the  decimal  .625  is  equal  to  f , 

8)  5.0  0  0         may  be  shown  by  writing  it  in  a  vulgar 
.625         fraction  and  reducing   it  thus,   T6^Vo  == 
f  Ans. 

NOTE.  It  is  also  evident,  that  .625  is  equal  to  f,  because  the 
numerators  have  equal  ratios  to  their  denominators. 

Hence  the  following 

RULE. 

Divide  the  numerator  ly  the  denominator,  annexing  one 
or  more  ciphers  to  the  numerator,  and  the  quotient  will  be 
the  decimal  required. 

NOTE.  It  is  not  usually  necessary,  that  decimals  should  be  carried 
to  more  than  six  places. 

2.  Reduce  f  to  a  decimal.  Ans.  .75. 

3.  Reduce  J  to  a  decimal.  Ans.  .875. 

4.  What  decimal  fraction  is  equal  to  T\  ?     Ans.  .4375. 

5.  Reduce  T4T  to  a  decimal.  Ans.  .363636+. 

6.  Reduce  ^  to  a  decimal.  Ans.  .416666 -f- 

II.  Reduce  compound  numbers  to  decimals. 

7.  Reduce  8s.  6d.  3qr.  to  the  decimal  of  a  £. 


SECT.31.]  DECIMAL    FRACTIONS.  H5 


4 

12 
20 


'RATION.  The  3  farthings  are  f  of  a  penny, 

3.00  and  these,  reduced  to  decimals,   are 

6.7  5  .75  of  a  penny,  which  we  annex  to 

8.5  625  the  pence,  and  proceed  in  the  same 


428125      manner  with  the  other  terms. 
Hence  the  following 

RULE. 

Write  the  given  numbers  perpendicularly  under  each 
other  for  dividends,  proceeding  orderly  from  the  least  to 
the  greatest ;  opposite  to  each  dividend  on  the  LEFT  hand, 
place  such  a  number  for  a  divisor,  as  will  bring  it  to  the 
next  superior  name,  and  draw  a  line  between  them.  Begin 
at  the  highest,  and  write  the  quotient  of  each  division,  as 
decimal  parts,  on  the  RIGHT  of  the  dividend  next  below  it, 
and  so  on,  until  they  are  all  divided ;  and  the  last  quotient 
will  be  the  decimal  required. 

8.  Reduce  15s.  6d.  to  the  fraction  of  a  £.      Ans.  .775. 

9.  Reduce  5cwt.  2qr.  141b.  to  the  decimal  of  a  ton. 

Ans.  .28125. 

10.  Reduce  3qr.  211b.  to  the  decimal  of  a  cwt. 

Ans.  .9375. 

11.  Reduce  6fur.  8rd.  to  the  decimal  of  a  mile. 

Ans.  .775. 

12.  Reduce  3R.  19p.  167ft.  72in.  to  the  decimal  of  an 
acre.  Ans.  .872595-}-. 

NOTE  1.  If  it  be  required  to  reduce  pounds,  shillings,  pence,  and 
farthings,  of  the  old  New  England  currency,  to  dollars,  cents,  and 
mills  ;  the  pounds,  shillings,  &c.  may  be  reduced  to  the  decimal  of  a 
£ ;  and  if  this  decimal  be  multiplied  by  10  and  the  product  divided 
by  3,  the  quotient  will  be  dollars  and  cents.  But  if  the  above  deci 
mal  be  multiplied  by  10,  and  the  product  be  divided  by  4,  the  quotient 
will  be  the  reduction  of  the  old  currency  of  New  York  to  dollars  and 
cents. 

NOTE  2.  If  it  be  required  to  bring  English  sterling  money  to  dol 
lars  and  cents,  let  the  pounds,  &c.  be  reduced  to  the  decimal  of  a 
penny  ;  then  divide  this  decimal  by  j ,  and  the  quotient  is  dollars 
and  cents. 

13.  Change  18£.  15s.  6d.  of  the  old  New  England  cur 
rency,  to  dollars  and  cents. 


116  DECIMAL    FRACTIONS.  [SECT.  31. 

OPERATION. 

18JB.  15s.  6d.=  lS.775£.;  18.775  x  *£-  =  8  62.58|  Ans. 

14.  Change  15£.   15s.  9d.  of  the  old  currency  of  New 
York,  to  dollars  and  cents. 

OPERATION. 

15£.  15s.  9d.=  15.7875£.;  15.7875 X^°=$ 39.46.8 j  Ans. 

15.  Change    176c£.    19s.    9d.    sterling  to  United    States 
currency.  Ans.  $  786.61  -)-. 

OPERATION. 

176£.  19s.  9d.  =176.9875<f .;  176.9876 xV=$  786.61  +. 

III.   To    find   the  value  of  any  given    decimal  in  the 
terms  of  the  integer. 

16.  What  is  the  value  of  .9S75£.  ?  Ans.  19s.  9d. 

OPERATION. 

.9875  This   question  is  performed  by 

2  0  the    same    principle   we    adopted 

197500  *n  finding  the  value   of  a  vulgar 

I  2  fraction  in  the  known  parts  of  the 


9.0000 
Hence  the  following 

RULE. 

Multiply  the  given  decimal  by  that  number  which  it  takes 
of  the  next  denomination  to  make  one  of  that  greater,  and 
cut  off  as  many  places  for  a  REMAINDER,  on  the  RIGHT  hand, 
as  there  are  places  in  the  given  decimal.  Multiply  the  RE 
MAINDER  by  the  next  lower  denomination,  and  cut  off  for  a 
remainder  as  before,  and  so  proceed,  until  the  decimal  is 
reduced  to  the  denomination  required ;  the  several  denomi 
nations  standing  at  the  LEFT  hand  are  the  answers  required. 

1.  What  is  the  value  of  .628125  of  a  £  ? 

Ans.  12s.  6}d. 

2.  What  is  the  value  of  .778125  of  a  ton  ? 

Ans.  15cwt.  2qr.  71b. 

3.  What  is  the  value  of  .75  of  an  ell  English  ? 

Ans.  3qr.  3na. 


SECT.  32.]  DECIMAL   FRACTIONS.  117 

4.  What  is  the  value  of  .965625  of  a  mile  ? 

Ans.  7fur.  29rd. 

5.  What  is  the  value  of  .94375  of  an  acre  ? 

Ans.  3R.  31p. 

6.  What  is  the  value  of  .815625  of  a  pound  Troy  ? 

Ans.  9oz.  lodwt.  18gr. 

7.  What  is  the  value  of  .5555  of  a  pound  apothecary's 
weight  ?  Ans.  6§. 53.  09. 


Section 

EXERCISES  IN  DECIMALS. 

1.  What  is  the  value  of  locwt.  3qr.  141b.  of  coffee  at 
$9.50  per  cwt.  ?  Ans.  $  150.8 1-f. 

2.  What  cost  17T.  IScwt.  Iqr.  71b.  of  potash  at  $53.80 
per  ton  ?  Ans.  $  963.86+. 

3.  What  cost  37A.   3R.    16p.    of  land    at    $75.16    per 
acre  ?  Ans.  $2844.80+. 

4.  What   cost    15yd.  3qr.    2na.    of    cloth    at    $  3.75  per 
yard  ?  Ans.  $  59.53+ 

5.  What  cost  15|  cords  of  wood  at  $  4.62|  per  cord  ? 

Ans.  $71.10+. 

6.  What  cost   the    construction  of  17m.   6fur.  36rd.   of 
railroad  at  $3765.60  per  mile  ?        Ans.  $67263.03+ 

7.  What    cost    27hhd.    21gal.    of    temperance    wine    at 
$  15.371  per  hogshead  ?  Ans.  $  420.24+. 

8.  What   are  the  contents  of  a  pile  of  wood,  18ft.  9in. 
long,  4ft.  6in.  wide,  and  7ft.  3in.  high  ? 

Ans.  611ft.  1242in. 

9.  What  are  the  contents  of  a  board  12ft.  6in.  long,  and 
2ft.  9in  wide  ?  Ans.  34ft.  54in. 

10.  Bought  a  cask  of  vinegar  containing  25gal.  3qt.  Ipt. 
at  $  0.37|-  per  gallon  ;  what  was  the  amount  ? 

Ans.  $9.70+. 

11.  Bought  a  farm  containing  144A.  3R.  30p.  at  $97.62J 
per  acre  ;   what  was  the  cost  of  the  farm  ? 

Ans.  $  14149.52+. 


]18  SIMPLE    INTEREST.  [SECT.  33. 

12.  Sold  Joseph  Punson  3T.  18cwt.  211b.  of  salt  hay, 
at  $9.37£  per  ton.     He  having  paid  me  $20.25,  what 
remains  due  ?  Ans.  $  16.40-}-. 

13.  If  J  of  a  cord  of  wood  cost  $  5.50,  what  cost  one 
cord  ?     What  cost  7J  cords  ?  Ans.  $  48.71+. 


Section  33. 

SIMPLE     INTEREST. 

INTEREST  is  the  compensation,  which  the  borrower  of 
money  makes  to  the  lender. 

PRINCIPAL  is  the  sum  lent. 

AMOUNT  is  the  interest  added  to  the  principal. 

PER  CENT.,  a  contraction  of  per  centum,  is  the  rate 
established  by  law,  or  that  which  is  agreed  on  by  the 
parties,  and  is  so  much  for  a  hundred  dollars  for  one 
year. 

GENERAL  RULE. 

Let  the  per  cent,  "be  considered  as  a  decimal  of  a  hun 
dred  dollars,  and  multiply  the  principal  by  it,  and  the  pro 
duct  is  the  interest  for  one  year ;  but,  if  it  be  required  to 
Jind  the  interest  for  more  than  one  year,  multiply  the  pro 
duct  by  the  number  of  years. 

NOTE.  The  decimal  for  6  per  cent,  is  .06  ;  for  7  per  cent.  .07  ;  for 
8  per  cent.  .08 ;  for  9|  per  cent.  .0925 ;  for  2£  per  cent.  .025,  &c. 
The  decimals  must  be  pointed  off  as  in  Multiplication  of  Decimal 
Fractions. 

This  rule  is  obvious  from  the  fact,  that  the  rate  per 
cent,  is  such  a  part  of  every  hundred  dollars.  Thus,  6 
per  cent,  is  pf^  of  the  principal. 

NOTE.  When  no  particular  per  cent,  is  named,  6  per  cent,  is  to  be 
understood,  as  it  is  the  legal  interest  in  the  New  England  States  gen 
erally.  In  New  York  the  legal  interest  is  7  per  cent. 


SECT.  33.]  SIMPLE    INTEREST.  119 

1.   What  is  the  interest  of  $  346  for  one  year  ? 

Ans.  $  20.76. 

OPERATION. 

3  4  Q  There  being  two  places  of  deci- 

0  Q  mals  in  the  multiplier,  we  point  off 

tw°  in  the  product 


2.  What  is  the  interest  of  $  67.87  for  5  years  ? 

Ans.  $  20.36. 

OPERATION. 

'Q  ~  There  being  two  places  of  deci- 

-  1  —  mals  in  the  multiplicand,  and  two  in 

4.0  722  tne    multiplier,    we    point    off   four 

_  5  places  in  the  product. 

$20.3610 

3.  What  is  the  interest  of  8  197  for  1  year  ? 

Ans.  $  11.82. 

4.  What  is  the  interest  of  $  1728  for  3  years  ? 

Ans.  $311.04. 

5.  What  is  the  interest  of  $  69  for  2  years  ? 

Ans.  $  8.28. 

6.  What  is  the  interest  of  $  1775  for  7  years  ? 

Ans.  $  745.50. 

7.  What  is  the  interest  of  $  987  for  10  years  ? 

Ans.  $  592.20. 

8.  Required  the  interest  of  $69.17  for  4  years. 

Ans.  $  16.60. 

9.  Required  the  interest  of  $  96.87  for  11  years. 

Ans.  $63.93. 

10.  Required  the  interest  of  $  1.95  for  18  years. 

Ans.  $2.10. 

11.  Required  the  interest  of  $  1789  for  20  years. 

Ans.  $2146.80. 

12.  Required  the  interest  of  $  666.66  for  30  years. 

Ans.  $1199.98. 

13.  What  is  the  amount  of  $  98.50  for  5  years  ? 

Ans.  $  128.05. 

14.  What  is  the  amount  of  $  168.13  for  11  years  ? 

Ans.  $  279.09. 

15.  What  is  the  amount  of  $75.75  for  17  years  ? 

Ans.  $153.01. 

16.  Required  the  amount  of  $675.50  for  100  years. 

Ans.  $  4723.50. 


120  SIMPLE    INTEREST.  [SECT.  33. 

II.  To  find  the  interest  for  months,  at  six  per  cent. 

RULE. 

Multiply  the  principal  by  half  the  number  of  months,  ex 
pressed  decimally  as  a  per  cent. ;  that  is,  for  12  months 
multiply  by  .06 ;  for  8  months  multiply  by  .04 ;  for  7 
months  .035 ;  for  1  month  .005,  and  point  for  decimals  as 
in  the  last  rule. 

NOTE.  It  is  obvious,  that  if  half  the  number  of  months  were  12,  it 
would  be  1  per  cent,  a  month,  that  is,  half  the  months  will  be  the  per 
cent.  Q.  e.  d. 

1.  What  is  the  interest  of  $  486  for  10  months  ? 

OPERATION. 

486  principal. 
.0  5  months  decimal  of  the  per  cent. 

$~2T.3~0  Ans. 

2.  What  is  the  interest  of  $  1728  for  18  months  ? 

Ans.  $  155.52. 

3.  What  is  the  interest  of  $  16.87  for  20  months  ? 

Ans.  $  1.68. 

4.  Required  the  interest  of  $  118.15  for  30  months. 

Ans.  $  17.72. 

5.  Required  the  interest  of  $97.16  for  17  months. 

Ans.  $  8.25. 

6.  Required  the  interest  of  $  789.87  for  23  months. 

Ans.  $  90.83. 

7.  Required  the  amount  of  $978.18  for  27  months. 

Ans.  $1110.23. 

8.  Required  the  amount  of  $87.96  for  1  month. 

Ans.  $88.39. 

9.  Required  the  amount  of  $  81.81  for  100  months. 

Ans.  $  122.71. 

10.  Required  the  amount  of  $  0.87  for  87  months. 

Ans.  $  1.24. 

III.  To  find  the  interest  for  any  sum  for  months  and 
days,  at  6  per  cent. 

RULE. 

To  one  half  of  the  months  expressed  decimally  as  in  the 
last  rule,  annex  one  sixth  of  the  days.  With  this  multiply 


SECT.  33.]  SIMPLE   INTEREST.  121 

the  principal,  and  point  off  in  the  product  as  many  deci 
mals  as  there  are  in  loth  factors  ;  the  first  two  figures  at 
the  right  of  the  separatrix  are  cents,  and  the  third  is  mills. 

NOTE.  If  any  other  per  cent,  is  required,  proceed  as  before,  and 
then  divide  the  product  by  6,  and  multiply  the  quotient  by  the  rate 
required.  The  same  result  will  be  obtained  if  we  multiply  by  the  re 
quired  rate,  and  divide  the  product  by  6. 

1.  What  is  the  interest  of  $  57.50  for  10  months  and  24 
days  ?  Ans.  $3.105. 

OPERATION. 

5  0*5  4  We  multiP17  by  -054>  because  .05 

is  the  rate  per  cent,  for  10  months  ; 
23000  anc|  we  annex  the  4,  because  4  is  4- 

2j^50_  of  the  24  days. 

3.10500 

2.  What  is  the  interest  of  $  178.75  for  17  months  17  days 
at  7  per  cent.  ?  Ans.  $  18.31. 

OPERATION. 

178.75 


125125 
143000 
14895 

6)  1570020 

261  670 

___  7 

1  8.3  1  6  9  0 

3.  What  is  the  interest  of  $761.75  for  14  months  and 
18  days  ?  Ans.  $55.60. 

4.  What  is  the  interest  of  $1728.19  for  17  months  and 
10  days  ?  Ans.  $  149.77. 

5.  What  is  the  interest  of  $88.96  for  16  months  6  days  ? 

Ans.  $7.20. 

6.  What  is  the  interest  of  $  107.50  for  1  month  29  days  ? 

Ans.  $1.05. 

7.  What  is  the  interest  of  $87.25  for  20  months  5  days  ? 

Ans.  $8.79. 

8.  What  is  the  interest  of  $  73.16  for  19  months  23  days? 

Ans.  $7/J3. 


122  SIMPLE   INTEREST.  [SECT.  33. 

9.  What  is  the  interest  of  $  1.71  for  24  months  2  days  ? 

Ans.  $  0.20. 

10.  Required  the  interest  of  $  100  for  100  months  1  day. 

Ans.  $50.01. 

11.  Required  the  interest  of  three  dollars  and  five  cents 
for  2  months  and  2  days.  Ans.  $  0.03. 

IV.  When  the  interest  is  required  on  any  sum,  from  a 
certain  day  of  the  month  in  a  year,  to  a  particular  day  of 
a  month  in  the  same,  or  in  another  year,  we  adopt  the 
following 

RULE. 

Find  the  time  by  placing  the  latest  date  in  the  upper  line, 
and  the  earliest  date  under  it.  Let  the  year  be  placed  Jirst ; 
and  the  number  of  months  that  have  elapsed  since  the  year 
commenced  annexed  at  the  right  hand,  and  the  day  of  the 
month  next ;  then  subtract  the  earlier  from  the  later  date, 
and  the  remainder  is  the  time,  for  which  the  interest  is 
required.  Then  proceed  as  in  the  last  rule. 

NOTE.  Some  Arithmeticians  prefer  reckoning  the  months  by  their 
ordinal  number,  as  in  operation  £d. 

1.  What  is  the  interest  of  $  172.50,  from  Sept.  25,  1840, 
to  July  9,  1842  ?  Ans.  $  18.51.5. 

OPERATION    1st.  OPERATION   2d. 

Y.    mo.   da.  Y.    mo.   da. 

1842  6   9  1842  7   9 

1840  8  25  1840  9  25 

1  9  14  1  9  14 

$172.50  It  will  be  perceived,  that  the 

1.0  7£  result  in  finding  the  time  is  the 

120750  same  in  operation  2d,  as  in  ope- 

17250  ration  1st. 

5750 
$  1  8.5  1  5  0  0 

2.  What  is  the  interest  of  $  169.75,  from  Dec.  10,  1838, 
to  May  5,  1841  ?  Ans.  $24.47. 

3.  What  is  the  interest  of  $  17.18,  from  July  29,   1837, 
to  Sept.  1,  1841  ?  Ans.  $4.21. 


SECT.33.]  SIMPLE    INTEREST.  123 

4.  What  is  the  interest  of  $  67.07,  from  April  7,   1839, 
to  Dec.  11,  1841  ?  Ans.  $10.77. 

5.  Required  the  interest  of  $  117.75,  from  Jan.  7,  1839, 
to  Dec.  19,  1841.  Ans.  820.84. 

6.  Required  the  interest  of  $847.15,  from  Oct.  9,  1839, 
to  Jan.  11,  1843.  Ans.  $  165.47. 

7.  Required  the  interest  of  $7.18,  from  March  1,  1841, 
to  Feb.  11,  1842.  Ans.  $  0.40. 

8.  What  is  the  interest  of  $  976.18,  from  May  29,  1842, 
to  Nov.  25,  1845  ?  Ans.  $204.34. 

9.  I  have  John  Smith's  note  for  $  144,  dated  July  25, 
1839  ;   what  is  due  March  9,  1842  ?       Ans.  $  166.65. 

10.  L.  Johnson  has   J.  Kimball's   note,  dated   June  4, 
1841,  for  $  123  ;  what  is  due  to  Johnson  Dec.  7,  1843  ? 

Ans.  $141.51. 

11.  George  Cogswell    has    two    notes  against   J.  Doe  ; 
the  first  is   for    $375.83,   and  is  dated  Jan.  19,  1840  ; 
the  other  is  for  $76.19,  dated  April  23,  1841  ;   what  is 
the  amount  of  both  notes  Jan.  1,  1842  ? 

Ans.  $499.14. 

12.  What  is  the  interest  of  $68.19,  at  7  per  cent.,  from 
June  5,  1840,  to  June  11,  1841  ?  Ans.  $  4.85. 

13.  Required  the  amount  of  $79.15,  from  Feb.  17,  1839, 
to  Dec.  30,  1842,  at  7J  per  cent.  Ans.  $  102.11. 

14.  What  is  the  amount  of  $89.96,  from  June  19,  1840, 
to  Dec.  9,  1841,  at  8J  per  cent.  Ans.  $  100.88. 

15.  A.  Atwood    has   J.    Smith's   note    for    $  325,  dated 
June  5,   1839  ;    what  is  due  at  7^    per    cent.,  July  4, 
1841  ?  Ans.  $374.02. 

16.  J.  Ayer  has  D.  How's  note  for  $  1728,  dated  Dec. 
29,  1839  ;    what  is  the  amount  Oct.  9,   1842,   at  9  per 
cent.  ?  Ans.  $2160.00. 

17.  What  is  the  interest  of  $976.18,  from  Jan.  29,  1841, 
to  July  4,  1842,  at  12  per  cent.  ?  Ans.  $  167.57. 

18.  What  is  the  interest  of  $  176.17,  from  June  19,  1839, 
to  Sept.  7,  1843,  at  9J  per  cent.  ?  Ans.  $  72.42. 

19.  What  is  the  amount  of  J.  Turner's  note  for  $87.25, 
dated  June  1,  1841,  to  Dec.  17,  1843,  at  5  per  cent.  ? 

Ans.  $98.35. 

20.  What  is  the  amount  of  $379.78,  from  Dec.  3,  1808, 
to  August  23,  1847,  at  7|  per  cent.  ?     Ans.  $  1519.48. 


124  SIMPLE   INTEREST.  [SECT.  34, 

Section  34. 

PARTIAL    PAYMENTS. 

I.  When  notes  are  paid  within  one  year  from  the  time 
they  become  due,  it  has  been  the  usual  custom  to  find 
the  amount  of  the  principal  from  the  time  it  became  due 
until  the  time  of  payment,  and  to  find  the  amount  of  each 
indorsement  from  the  time  it  was  paid  until  settlement, 
and  to  subtract  their  sum  from  the  amount  of  the  prin 
cipal. 

1.     $  1234.  Boston,  Jan.  1,  1843. 

For  value  received,  I  promise  to  pay  John  Smith,  or 
order,  on  demand,  one  thousand  two  hundred  thirty-four 
dollars,  with  interest.  John  Y.  Jones. 

Attest,  Samuel  Emerson. 

On  this  note  are  the  following  indorsements. 
March  1,  1843.     Received  ninety-eight  dollars. 
June  7,  1843.     Received  five  hundred  dollars. 
Sept.  25,  1843.     Received  two  hundred  ninety  dollars. 
Dec.  8,  1843.     Received  one  hundred  dollars. 

What  remains  due  at  the  time  of  payment,  Jan.  1, 
1844  ?  Ans.  $293.12. 

Principal  $  1234.00 

Interest  for  one  year.  74.04 

Amount  1308.04 

First  payment  $  98.00 

Interest  for  10  months  4.90 

Second  payment  500.00 

Interest  for  6  months  24  days  17.00 

Third  payment  290.00 

Interest  for  3  months  6  days  4.64 

Fourth  payment  100.00 

Interest  for  23  days  38 

~$  1014.92 

Balance,  remains  due,  Jan.  1,  1844  $293.12 


SECT.  34.]  SIMPLE    INTEREST.  125 

«.     $  876.50.  Boston,  Sept.  25,  1842. 

For  value  received,  I  promise  to  pay  James  Savage,  or 
order,  on  demand,  eight  hundred  seventy-six  dollars  fifty 
cents,  with  interest.  Savage  James. 

Attest,  John  True. 

On  this  note  are  the  following  indorsements. 
Dec.  6,  1842.     Received  ninety-seven  dollars. 
Jan.  1,  1843.     Received  two  hundred  sixty-five  dollars. 
March  11,  1843.     Received  one  hundred  seventy  dollars. 
July  4,  1843.     Received  seventy-nine  dollars. 

What  remains  due  Aug.  6,  1843  ?         Ans.  $293.04. 

3.  $987.75.  Danvers,  Jan.  11,  1842. 
For  value  received,  we  jointly  and  severally  promise 

to  pay  Fitch  Pool,  or  order,  on  demand  two  months  from 
date,  nine  hundred  eighty-seven  dollars  seventy-five 
cents,  with  interest  after  two  months. 

John  T.  Johnson. 
Attest,  Isaiah  Webster.  Samuel  Jones. 

On  this  note  are  the  following  indorsements. 
May  1,  1842.     Received  three  hundred  dollars. 
June  5,  1842.     Received  four  hundred  dollars. 
Sept.  25,  1842.     Received  one  hundred  and  fifty  dollars. 

What  is  due  Dec.  13,  1842  ?  Ans.  $  156.94. 

4.  8800.  Bradford,  July  4,  1842. 
For  value  received,  I  promise  to  pay  Leonard  Johnson, 

or   order,    on   demand,    eight   hundred    dollars,  with  in 
terest.  Samuel  Neverpay. 
Attest,  Enoch  True. 

On  this  note  are  the  following  indorsements. 
Aug.  10,  1842.     Received  one  hundred  forty-four  dollars. 
Nov.  1,  1842.     Received  ninety  dollars. 
Jan.  I,  1843.     Received  four  hundred  dollars. 
March  4,  1843.     Received  one  hundred  dollars. 

What  remains  due  June  1,  1843  ?  Ans.  $88.02. 


126  SIMPLE   INTEREST.  [SECT.  34. 

II.  In  the  United  States'  Court,  and  in  most  of  the 
Courts  of  the  several  States,  the  following  rule  is  adopt 
ed  for  estimating  the  interest  on  notes  and  bonds,  when 
partial  payments  have  been  made. 

RULE. 

Compute  the  interest  on  the  principal  sum,  from  the  time 
when  the  interest  commenced  to  the  time  when  the  Jlrst  pay 
ment  icas  made,  which  exceeds,  either  alone  or  in  conjunc 
tion  with  the  preceding  payments,  if  any,  the  interest  at 
that  time  due  ;  add  that  interest  to  the  principal,  and  from 
the  sum  subtract  the  payment  made  at  that  time,  together 
with  the  preceding  payments,  if  any,  and  the  remainder 
forms  a  new  principal ;  on  which  compute  and  subtract  the 
interest,  as  upon  the  first  principal,  and  proceed  in  the 
same  manner  to  the  time  of  judgment. 

This  rule  is  illustrated  in  the  following  question. 
1.     $365.50.  Lynn,  Jan   1,  1842. 

For  value  received,  I  promise  to  pay  John  Dow,  or 
order,  on  demand,  three  hundred  sixty-five  dollars  fifty 
cents,  with  interest.  John  Smith. 

Attest,  Samuel  Webster. 

On  this  note  are  the  following  indorsements. 
June  10,  1842.     Received  fifty  dollars. 
Dec.  8,  1842.     Received  thirty  dollars. 
Sept.  25,  1843.     Received  sixty  dollars. 
July  4,  1844.     Received  ninety  dollars. 
Aug.  1,  1845.     Received  ten  dollars. 
Dec.  2,  1845.     Received  one  hundred  dollars. 

What  remains  due  Jan.  7,  1847  ?  Ans.  $  92.53. 

OPERATION. 

Principal  carrying  interest  from  Jan.  1,  1842,  to 

June  10,  1842  $305.50 

Interest    from   Jan.   1,   1842,   to   June    10,   1842, 

5  months  9  days  9.T>8 

Amount    375.18 

First  payment,  June  10,  1842  50.00 

Balance  for  new  principal  325.18 


SECT.31]  SIMPLE    INTEREST.  127 

Balance  for  new  principal  (brought  over)  325.18 

Interest  from  June  10,  1842,  to  Dec.  8,  1842, 

5  months  28  days  964 

Amount  334.  &2 

Second  payment,  Dec.  8,  1842  30.00 

Balance  for  new  principal  3U4.82 

Interest  from  Dec.  8,  1842,  to  Sept.  25,  1843, 

9  months  17  days  14.58 

Amount  319.40 

Third  payment,  Sept.  25,  1843  60.00 

Balance  for  new  principal  259.40 

Interest  from  Sept.  25,  1843,  to  July  4,  1844, 

9  months  9  days  12.06 

Amount  271.46 

Fourth  payment,  July  4,  1844  90.00 

Balance  for  new  principal  181. 4C 

Interest  from  July  4,  1844,  to  Dec.  2,  1845, 

16  months  28  days  15.36 

Amount     196.82 

Fifth  payment,  Aug.  1,  1845,  { •  riS^"  j     $  10-00 
Sixth  payment,  Dec.  2,  1845,  j*"£tt£f""|    100.00 

"110.00 

Balance  for  new  principal  86.82 

Interest    from    Dec.   2,    1845,   to   Jan.    7,    1847, 

13  months  5  days  5.71 

Remains  due  Jan.  7,  1847  $  92.53 

2.     $  1000.  Bradford,  Jan.  10,  1836. 

For  value  received,  I  promise  to  pay  James  Jones,  or 
order,  on  demand  with  interest  after  three  months,  one 
thousand  dollars.  John  Snow. 

Attest,  L.  True. 

On  this  note  are  the  following  indorsements. 
July  4,  1836.     Received  one  hundred  dollars. 
Jan.  1,  1837.     Received  two  hundred  dollars. 
Sept.  25,  1838.     Received  three  hundred  dollars. 
March  9,  1839.     Received  one  hundred  dollars. 
April  7,  1840.     Received  two  hundred  and  fifty  dollars. 

What  is  due  Jan.  10,  1842  ?  Ans.  $232.26. 


128  COMMISSION    AJ\D    BROKERAGE.      [SECT.  35. 

3.     $  1G66.  Newburyport,  June  5,  1838. 

For  value  received,  I  promise  to  pay  John  Boardman, 
or  order,  on  demand,  one  thousand  six  hundred  sixty-six 
dollars  with  interest.  John  J.  Fortune. 

Attest,  T.  Webster. 

On  this  note  are  the  following  indorsements. 
July  4,  1839.     Received  one  hundred  dollars. 
Jan.  1,  1840.     Received  ten  dollars. 
July  4,  1840.     Received  fifteen  dollars. 
Jan.  1,  1841.     Received  five  hundred  dollars. 
Feb.  7,  1842.     Received  six  hundred  fifty-six  dollars. 

What  is  due  Jan.  1,  1843  ?  Ans.  $  767.08. 


Section  35. 

COMMISSION    AND    BROKERAGE. 

COMMISSION  and  BROKERAGE  are  compensations  made 
to  factors,  brokers,  and  other  agents,  for  their  services, 
either  for  buying  or  selling  goods. 

NOTE.  A  factor  is  an  agent,  employed  by  merchants  residing  in 
other  places,  to  buy,  and  sell,  and  to  transact  business  on  their  ac 
count.  A  broker  is  an  agent  employed  by  merchants  to  transact 
business. 

RULE. 

The  questions  are  performed  in  the  same  manner  as  in 
interest. 

1.  What  is  the  commission  on  the  sale  of  $5678  value 
of  cotton  goods,  at  3  per  cent.  ?  Ans.  $  170.34. 

2.  A  broker  sells  goods  to  the  amount  of  $  7896,   at  2 
per  cent.,  what  is  his  commission  ?          Ans.  $  157.92. 

3.  My  agent  in  Lowell  has   purchased  goods  for  me  to 
the  amount  of  $  1728,  what  is  his  commission,  at  1 J  per 
cent.  ?  Ans.  625.92. 

4.  My  factor  advises  me,  that  he  has  purchased,  on  my 


SECT.  36.]  INSURANCE    AND    POLICIES.  129 

account,  97  bales  of  cloth,  at  $  15.50  per  bale  ;  what  is 
his  commission,  at  2|  per  cent.  ?  Ans.  $  37.58-J-. 

5.  My  agent,  at  New  Orleans,   informs  me,  that  he  has 
disposed  of   500  barrels  of  flour  at  $  6.50  per  barrel, 
88  barrels  of  apples  at  $2.75  per  barrel,  and  56  cwt.  of 
cheese  at  $  10.60  per  cwt.  ?  what  is  his  commission,  at 
3J  per  cent.  ?  Ans.  $  153.21. 

NOTE.  To  estimate  the  duties  on  imported  goods  is  performed  in 
the  same  manner  as  interest,  except  when  the  duties  are  so  much 
per  ton,  yard,  &c. 

6.  What  is  the  duty  on  $  8000  value  of  imported  goods, 
at  20  per  cent.  ?  Ans.  $  1600. 

7.  What    is   the  duty  on  50  tons   of  iron,  at    $  30  per 
ton  ?  Ans.  $  1500. 


Section   36. 

INSURANCE    AND    POLICIES. 

INSURANCE  is  a  security,  by  paying  a  certain  sum  to 
indemnify  the  secured  against  such  losses,  as  shall  be 
specified  in  the  policy. 

Policy  is  the  name  of  the  writ,  or  instrument,  by 
which  the  contract  or  indemnity  is  effected  between  the 
parties. 

RULE. 
The  same  as  in  interest. 

1.  What   is   the  premium  on   $  868,  at  12   per   cent.  ? 

Ans.  $  104.16. 

2.  What  is  the   premium  on    $  1728,  at   15  per  cent.  ? 

Ans.  $25.92. 

3.  A  house,  valued  at  $3500,  is  insured  at  If  per  cent.; 
what  is  the  premium  ?  Ans.   $61.25. 

4.  A  vessel  and  cargo,  valued  at  $  35000,  is  insured  at 
3|   per  cent.  ;  now,  if  this  vessel  should   be  destroyed, 
what  will  be  the  actual  loss  to  the  insurance  company  ? 

Ans.  $  33687.50. 


130  STOCKS.    BANKING.  [SECT.  37, 38. 

Section    37. 

STOCKS. 

STOCKS  is  the  general  name  used  for  funds,  established 
by  government  or  individuals,  in  their  corporate  capacity, 
the  value  of  which  is  often  variable. 

The  method  for  computation  is  the  same  as  in  interest. 

1.  What  must  be  given  for  10  shares  in  the  Boston  and 
Portland    Railroad,    at    15   per    cent,    advance,  shares 
being  $  100  each  ? 
$  100  X  10  =  $  1000  ;  $  1000  X  1.15  =  $  1150  Ans. 

£.  What  must  be  given  for  75  shares  in  the  Lowell 
Railroad,  at  25  per  cent,  advance,  the  original  shares 
being  $  100  each  ?  Ans.  $  9375. 

3.  What   is  the  purchase  of  $  8979  Bank  stock  at  12 
per  cent,  advance  ?  Ans.  $  10056.48. 

4.  What  is  the  purchase  of  $  1789  Bank  stock  at  9  per 
cent,  below  par  ?  Ans.  $  1627.99. 


Section  38. 

BANKING. 

When  a  note  is  discounted  at  a  bank,  the  interest  is 
taken  at  the  time  the  note  is  given,  and  the  interest  is 
computed  for  3  days  more  than  the  time  specified  in  the 
note  ;  that  is,  if  the  note  is  given  for  60  days,  the  in 
terest  is  taken  for  63  days  ;  for  the  law  allows  three  days 
to  the  debtor,  after  the  time  has  expired  for  payment, 
which  are  called  days  of  grace.  If,  therefore,  a  note  is 
given  to  the  President  and  Directors  of  the  Merrimack 
Bank  for  $  100,  to  be  paid  in  60  days,  the  interest  on  the 
$  100  is  computed  for  63,  and  taken  from  the  sum  of  the 
note.  So  that  the  borrower  receives  only  $  98.95  for  the 
note  discounted. 


SECT.39.]  DISCOUNT.  131 

1.  What  is  the  bank  discount  on  $  478,  for  60   days  ? 

Ans.  $5.01-|-. 

2.  What  is  the  bank  discount  on  $  780,  for  30  days  ? 

Ans.  $  4.29. 

3.  What  is  the  bank  discount  on  $  1728,  for  90  days  ? 

Ans.  $  26.78+. 

4.  How  much  money  should  be  received  on  a  note  of 
$  1000,  payable    in  4  months,    discounted    at    a   bank, 
where  the  interest  is  6  per  cent.  ?  Ans.  $  979.50. 


Section  3  9. 

DISCOUNT. 

The  object  of  discount  is,  to  show  what  allowance 
should  be  made,  when  any  sum  of  money  is  paid  before 
it  becomes  due. 

The  present  worth  of  any  sum  is  the  principal,  that 
must  be  put  at  interest,  to  amount  to  that  sum  in  the 
given  time.  That  is,  $  100  is  the  present  worth  of  $  106, 
due  one  year  hence  ;  because  $  100  at  6  per  cent,  will 
amount  to  8  106,  and  $  6  is  the  discount. 

Therefore  when  the  interest  is  6  per  cent,  the  present 
worth  is  J-£f  of  the  principal,  and  the  discount  is  T§^  of 
the  principal  ;  and  the  same  rule  will  hold  good  for  any 
other  per  cent. 

1.  What  is  the  present  worth  of  $  25.44,  due  one  year 
hence?  Ans.  $24.00. 

FIRST   METHOD.  SECOND    METHOD. 

2  5.4  4          1.0  6)  2  5.4  4  (  $  2  4  Ans. 
100  212 

106)2544(824  Ans.        424 
212  424 

424 
424 

From  the  above  illustration,  we  deduce  the  following 


132  COMPOUND   INTEREST.  [SECT.  40. 

RULE. 

Divide  the  given  sum  by  the  amount  of  $  1  for  the  given 
rate  and  time,  and  the  quotient  will  be  the  present  worth. 
Or,  multiply  the  given  sum  by  100,  and  divide  the  product 
by  the  amount  of  $  100  for  the  given  rate  and  time,  and  the 
quotient  is  the  present  worth. 

&.  What  is  the  present  worth  of  $  152.64,  due  one  year 
hence  ?  Ans.  $  144.00. 

3.  What  is  the  present  worth  of  $  477.71,  due  four  years 
hence  ?  Ans.  $  385.25. 

4.  What   is  the  present  worth  of  $  172.86,  due  3  years 
4  months  hence  ?  Ans.  $  144.05. 

5.  What  is  the  present  worth  of  $800,  due   3  years  7 
months  and  18  days  hence  ?  Ans   $656.81-}-. 

6.  Samuel  Heath  has  given  his  note  for   $  375.75,  dated 
Oct.  4,  1842,  payable  to  John  Smith,  or  order,  Jan.    1, 
1844  ;    what  is  the  real  value  of  the  note  at  the  time 
given  ?  Ans.  $  349.69+. 

7.  Bought  a  chaise    and   harness,  of   Isaac   Morse,  for 
$  125.75,  for  which  I  gave  him  my  note,  dated  Oct.  5, 
1842,  to  be  paid  in  six  months  ;    what   is   the    present 
value  of  the  note  Jan.  1,  1843  ?  Ans.  $  123.81+. 

8.  My  tailor  informs  me,  it  will  take  10  square  yards  of 
cloth   to  make   me   a  full  suit  of  clothes.     The  cloth  I 
am  about  to  purchase  is  If  yards  wide,  and  on  spunging 
it  will  shrink  5  per  cent,  in  width    and  length.      How 
many  yards  of  the  above  cloth  must  I  purchase  for  my 
"  new  suit  "  ?  Ans.  6yd.  Iqr. 


Section   4O. 

COMPOUND    INTEREST. 

The  law  specifies,  that  the  borrower  of  money  shall 
pay  a  certain  number  of  dollars,  called  per  cent.,  for  the 
use  of  $  100  for  a  year.  Now,  if  this  borrower  does  not 
pay  to  the  lender  this  per  cent,  at  the  end  of  the  year, 


SECT.  40.]  COMPOUND    INTEREST.  133 

it  is  no  more  than  just,  that  he  should  pay  interest  for 
the  use  of  it,  so  long  as  he  shall  keep  it  in  his  posses 
sion  ;  this  is  called  Compound  Interest. 

1.  What  is  the  compound  interest  of  $  500  for  3  years  ? 

Ans.  $  95.50. 
$500=  Principal. 

1.06 

30.00  =  Interest  for  1  year. 
500. 

530.00  =  Amount  for  1  year. 

JLL? 

3  1.80        =  Interest  for  second  year. 
530 

561.80        =  Amount  for  2  years. 
1.06 


3  3.7  0  8  0        =  Interest  for  third  year. 
561.80 

5  9  5.5  0.8  0        =  Amount  for  3  years. 
500 
$  9  5.5  0  =:  Compound  interest  for  3  years. 

From  the  above  process,  we  see  the  propriety  of  the 
following 

RULE. 

Find  the  interest  of  the  given  sum  for  one  year,  and  add 
it  to  the  principal ;  tlien  Jind  the  interest  of  this  amount  for 
the  next  year ;  and  so  continue,  until  the  time  of  settlement. 
Subtract  the  principal  from  the  last  amount,  and  the  re 
mainder  is  the  compound  interest. 

&.  What   is   the   compound   interest   of    $761.75   for  4 
years  ?  Ans.  $  199.94. 

3.  What  is  the  amount  of  $  67.25  for  3  years,  at  com 
pound  interest  ?  Ans.   $  80.09+. 

4.  What  is  the  amount  of  $  78.69  for  5  years  at  7  per 
cent.  ?  Ans.  8  110.33. 

5.  What  is  the  amount  of  $  128  for  3  years  5  months 
and  IcS  days,  at  compound  interest  ?       Ans.    8  15670. 

6.  What  is  the  compound  interest  of  $  76.18  for  2  years 
6  months  9  days  ?  Ans.  $  12.96. 

L 


134  COMPOUND    INTEREST.  [SECT.  40. 

II.  To  find  the  amount  of  a  note  at  compound  interest, 
when  there  have  been  partial  payments. 

RULE. 

Find  the  amount  of  the  principal,  and  from  it  subtract 
the  amount  of  the  indorsements. 

7.     $  144.  Haverhill,  Sept.  25,  1839. 

For  value  received,  I  promise  to  pay  Charles  North- 
end,  or  order,  on  demand,  one  hundred  forty-four  dollars, 
with  interest.  John  Small,  Jr. 

Attest,  Q.  Jones. 

On  this  note  are  the  following  indorsements. 

Jan.  1,  1840.     Received  thirty  dollars. 
June  30,  1841.     Received  eighty  dollars. 
Feb.  7,  1842.     Received  ten  dollars. 

What  is  due  on  the  above  note  at  compound  interest, 
Oct.  4,  1842  ?  Ans.  $  40.02. 

OPERATION   BY  COMPOUND    INTEREST. 

Principal  $  144.00 

Interest  from  Sept.  25,  1839,  to  Oct.  4,  1842  27.76 

Amount  171.76 

First  payment  $  30.00 

Interest  from  Jan.  1,  1840,  to  Oct.  4,  1842    5.23 
Second  payment  80.00 

Interest  from  June  30, 1841,  to  Oct.  4,  1842   6. 12 
Third  payment  10.00 

Interest  from  Feb.  7,  1842,  to  Oct.  4,  1842        39 

Amount  $131.74 
Remains  due,  Oct.  4,  1842  $  40.02 


SECT.  41.]  EQUATION    OF    PAYMENTS.  135 

Section   41. 

EQUATION    OF    PAYMENTS. 

When  several  sums  of  money,  to  be  paid  at  different 
times,  are  reduced  to  a  mean  time  for  the  payment  of 
the  whole,  without  gain  or  loss  to  the  debtor  or  creditor, 
it  is  called  Equation  of  Payments. 

1.  John  Jones  owes  Samuel  Gray  $  100  ;  $  20  of  which 
is  to  be  paid  in  2  months  ;  $  40  in  6  months  ;  $  30  in 
8  months  ;  and  $  10  in  12  months  ;  what  is  the  equated 
time  for  the  payment  of  the  whole  sum  ? 

Ans.  6mo.  12da. 

OPERATION.  By  analysis,   $20  for  2 

$20x2     —     40  months    is    the    same,   as 

$40x6r=240  $40   for    1    month  ;     and 

$30x8     =240  $40  for  6  months  is  the 

$10  x  1  2  =  120  same,     as     $  1     for    240 

$TOO       100)64  0(6  mo.       months  ;    and   $  30  for  8 

Q  0  0  months   is   the    same,    as 

-jfl  $  1  for  240  months  ;   and 

g  Q  $  10  for  12  months  is  the 

same,     as     $  1     for     120 

100)  1200  (12  da.     months;     therefore,     SI 

for    40  +  240  +  240  + 
120  =  640  months  is  the 

same,  as  $  20  for  2  months,  $  40  for  6  months,  $  30  for 
8  months,  and  $  10  for  12  months;  but  $20 +  $40  + 
$  30  +  $  10  are  $  100  ;  therefore,  $  1  for  640  months  is 
the  same,  as  $  100  for  y^  of  640  months,  which  is  6 
months  and  12  days,  as  before.  Hence  the  following 

RULE. 

Multiply  each  payment  by  the  time  at  which  it  is  due, 
then  divide  the  sum  of  the  products  by  the  sum  of  the  pay 
ments,  and  the  quotient  wiU  be  the  true  time  required. 

£.  John  Smith  owes  a  merchant,  m  Boston,  $  1000, 
$250  of  which  is  to  be  paid  in  4  months,  $350  in  8 


136  EQUATION    OF    PAYMENTS.  [SECT.  41. 

months,  and  the  remainder  in  12  months  ;  what  is  the 
equated  time  for  the  payment  of  the  whole  sum  ? 

Ans.  8mo.  18da. 

NOTE.  The  following  example  will  illustrate  the  method,  the 
merchants  practise  to  find  the  medium  time  of  payment  of  goods  sold 
on  credit. 

3.  Purchased  of  James  Brown,  at  sundry  times,  and  on 
various  terms  of  credit,  as   by  the  statement  annexed. 

When  is  the  medium  time  of  payment  ? 

Jan.       1,  a  bill  amounting  to  $  360,  on  3  months'  credit. 

Jan.     15,     do.           do.  186,  on  4  months'  credit. 

March   1,     do.           do.  450,  on  4  months'  credit. 

May     15,     do.           do.  300,  on  3  months'  credit. 

June   20,     do.           do.  500,  on  5  months'  credit. 

FORM    OF    STATEMENT. 

Due  April    1,  $360 

May  15,  $  1  8 6  x  45=  8370 
July  l,$450x  91=  40950 
Aug.  15,  $300x136=  40800 
Nov.  20,  $500x233=  116500 

1796  )  2  0  6  6  2  0  ( 1  1  5  /&  days. 

1796 
~~27~02 
1796 

~9~61To 

8980 
~80 

The  medium  time  of  payment  will  be  116  days  from 
April  1,  which  will  be  July  25. 

4.  Sold    S.  Dana  several    parcels   of  goods,  at  sundry 
times,  and  on  various  terms  of  credit,  as  by  the  follow 
ing  statement. 

Jan.     7,  1841,  a  bill  amounting  to  $  375.60,  on  4  months. 
Apr.  18,  1841,     do.         do.  687.25,  on  4  months. 

June    7,  1841,     do.         do.  568.50,  on  6  months. 

Sept. 25,  1841,     do.         do.  300.00,  on  6  months. 

Nov.    5,  1841,     do.         do.  675.75,  on  9  months. 

Dec.    1,1841,     do.         do.  100.00,  on  3  months. 

What  is  the  equated  time  for  payment  of  all  the  bills  ? 

Ans.  Dec.  24. 


SECT. 42.]  PROPORTION.  137 

Section   43. 

PROPORTION. 

PROPORTION  is  the  likeness  or  equalities  of  ratios. 
Thus,  because  4  has  the  same  ratio  to  8,  that  6  has  to 
12,  we  say  such  numbers  are  proportionals. 

If.  therefore,  any  four  numbers  whatever  be  taken,  the 
first  is  said  to  have  the  same  ratio  or  relation  to  the  sec 
ond,  that  the  third  has  to  the  fourth,  when  the  first  num 
ber,  or  term,  contains  the  second,  as  many  times,  as  the 
third  contains  the  fourth  ;  or,  when  the  second  contains 
the  first,  as  many  times,  as  the  fourth  does  tho  third. 
Thus,  9  has  the  same  ratio  to  3,  that  12  has  to  4,  because 
9  contains  3,  as  many  times,  as  12  does  4.  And  10  has 
the  same  ratio  to  5,  that  12  has  to  6,  because  10  contains 
5,  as  many  times,  as  12  does  6.  Ratios  are  represented 
by  colons  ;  and  equalities  of  ratios  by  double  colons. 

The  first  and  third  terms  are  called  antecedents,  and 
the  second  and  fourth  are  called  consequents ;  also,  the 
first  and  fourth  terms  are  called  extremes,  and  the  second 
and  third  are  called  means. 

Whatever  four  numbers  are  proportionals,  if  their 
antecedents  and  consequents  be  multiplied  or  divided  by 
the  same  numbers,  they  are  still  proportionals  ;  and,  if 
the  terms  of  one  proportion  be  multiplied  or  divided  by 
the  corresponding  terms  of  another  proportion,  their 
products  and  quotients  are  still  proportionals. 

If  the  product  of  the  extremes  be  equal  to  the  product 
of  the  means,  it  is  evident,  that  if  any  three  of  the  four 
proportionals  be  given,  the  other  may  be  obtained  ;  for, 
if  the  product  of  the  means  be  divided  by  one  of  the  ex 
tremes,  the  quotient  will  be  the  other  extreme  ;  and,  if 
the  product  of  the  extremes  be  divided  by  one  of  the 
means,  the  quotient  will  be  the  other  mean.  Hence  the 
following 

RULE. 

State  the  question  by  making  that  number ',  which  is  of  the 
same  name  or  quality  as  the  answer  required,  the  third  term ; 

L* 


138  PROPORTION.  [SECT.  42. 

then,  if  the  answer  required  is  to  be  greater  than  the  third 
term,  make  the  second  term  greater  than  the  first ;  but  if  the 
answer  is  to  be  less  than  the  third  term,  make  the  second 
less  than  the  Jirst. 

Reduce  the  Jirst  and  second  terms  to  the  lowest  denomina 
tion  mentioned  in  either,  and  the  third  term  to  the  loirest 
denomination  mentioned  in  it. 

Multiply  the  second  and  third  terms  together,  and  divide 
their  product  by  the  Jirst,  and  the  quotient  is  the  answer  in 
the  same  denomination  to  which  the  third  is  reduced. 

If  any  thing  remains,  after  division,  reduce  it  to  the  next 
lower  denomination,  and  divide  as  before. 

If  either  of  the  terms  consists  of  Jr actions,  state  the  ques 
tion  as  in  whole  numbers,  and  reduce  the  mixed  numbers  to 
improper  fractions,  compound  fractions  to  simple  ones,  and 
invert  the  Jirst  term,  and  then  multiply  the  three  terms  con- 
tinually  together,  and  the  product  is  the  answer  to  the  ques 
tion.  Or,  the  fractions  may  be  reduced  to  a  common  de 
nominator  ;  and  their  numerators  may  be  used  as  whole 
numbers.  For  when  fractions  are  reduced  to  a  common  de 
nominator,  their  value  is  as  their  numerators. 

NOTE  1.  It  may  be  observed  in  Proportion,  that  the  third  term  is 
the  quantity,  whose  price  or  value  is  wanted,  and  that  the  second 
term  is  the  value  of  the  first;  when,  therefore,  the  second  term  is 
multiplied  by  the  third,  the  product  is  as  much  more  than  the  answer, 
as  the  first  term  is  greater  than  unity  ;  therefore,  by  dividing  the  pro 
duct  by  the  first  term,  we  have  the  value  of  the  quantity  required. 

NOTE  2.  The  pupil  should  perform  every  question  by  analysis, 
previous  to  his  performing  it  by  Proportion. 

1.  If  71bs.  of  sugar  cost  56  cents,  what  cost  361bs.  ? 

ibs.     ibs.         ct3.  ln  stating  this  question,  we   make 

7   :  36   ::  56       56  cents  the  third  term,  because  the 

5  6  answer  will  be  in  cents.      And,  as 

216  we  perceive  from  the  nature  of  the 

180  question,  that  the  answer  or  fourth 

7  >\  2  0  l"fi  term  will  be  more  than  56  cents,  we 

'  t  _  know,  that  of  the  other  two  terms, 

$2.88  Ans.  ihe  second  must  be  larger  than  the 

frst.y  we  therefore  make  361bs.  the 

second  term,  and  71bs.  the  first  term. 


IECT.  42.]  PROPORTION.  139 

To  perform  this  question  by  analysis,  we  say.  If  Tibs, 
cost  56  cents,  one  Ib.  will  cost  |  of  56  cents,  which  are 
8  cents.  Then,  if  lib.  cost  8  cents,  361bs.  will  cost  36 
times  as  much  ;  that  is,  36  times  8  cents,  which  are 
8  2.88  Ans.  as  before. 

2.  If  76  barrels  of  flour  cost  $  456,  what  cost  12  barrels? 

**""•       tar'  As  the  answer  to  this  ques- 

76  :   1/2  : :  456        tion  w{\\  \^e  m  dollars,  we  place 

456  $  456  in  the  third  term  ;   and, 

72  as  the  answer  or   fourth   term 

6  0  must    be   less  than  $  456,   be- 

48  $                  cause  12  barrels  will  cost  less 

76)5472(72  Ans.      than  7G  barrels5  we  must,  of 

^32  the  other  two  terms,  make  the 

.  _  ~  less  the  second  term,  and  the 

}  ^  2  larger  the  first  term  ;  that  is,  12 

barrels  must  be  the  second  term, 

and  76  barrels  the  first  term. 

We  analyze  this  question  by  saying,  if  76  barrels  cost 
6456,  1  barrel  will  cost  TV  of  $456,  which  is  $6. 
Then,  if  1  barrel  cost  §6,  12  barrels  will  cost  12  times 
as  much,  that  is,  $  72  Ans.  as  before. 

3.  If  3  men  can  dig  a  well  in  20  days,  how  long  would  it 
take  12  men  ? 

men.      men.       days.  As   the    answer   will 

12  :  3  ::  20  be    in    days,    so    the 

third     term     will     be 

12)60(5  days,  Ans.      days.    As  12  men  will 
6  0  dig   the    well    in    less 

time     than     3      men, 
therefore,  the  second  term  will  be  less  than  the  first. 

By  analysis.  If  3  men  dig  the  well  in  20  days,  it  will 
take  one  man  3  times  as  long,  that  is,  60  days.  Again, 
we  say,  If  one  man  dig  the  well  in  60  days,  12  men 
would  dig  it  in  ^  of  60  days,  that  is,  5  days,  Ans.  as 
before. 

4.  If  41bs.  of  beef  cost  36  cents,  what  cost  871bs.  ? 

Ans.  $  7.83. 

5.  What   cost  9  gallons  of  molasses,  if  63  gallons  cost 
$14.49?  Ans.  $2.07 


140  PROPORTION.  [SECT.  42. 

6.  What  cost  97  acres  of  land,  if  19  acres  can  be  ob 
tained  for  $337.25  ?  Ans.  $  1721.75. 

7.  If  a  man  travel  319  miles  in  11  days,  how  far  will  he 
travel  in  47  days  ?  Ans.  1363  miles. 

8.  If  71bs.  of  beef  will  buy  41bs.  of  pork,  how  much  beef 
will  be  sufficient  to  buy  481bs.  of  pork  ?      Ans.  841bs. 

9.  Paid  for  87  tons  of  iron   $5437.50,  how  many  tons 
will  $  7687.50  buy  ?  Ans.  123  tons. 

10.  When    $  120  are  paid  for  15  barrels  of   mackerel, 
what  will  be  the  cost  of  79  barrels  ?  Ans.  $  632. 

11.  If  9  horses  eat  a  load  of  hay  in  12  days,  how  many 
horses  would  it  require  to  eat  the  hay  in  3  days  ? 

Ans.  36  horses. 

12.  When  $  5.88  are  paid  for  7  gallons  of  oil,  what  cost 
27  gallons  ?  Ans.  $  22.68. 

13.  When  $  10.80  are  paid  for  91bs.  of  tea,  what  cost 
1471bs.  ?  Ans.  $  176.40. 

14.  What  cost  27  tons  of  coal,  when  9  tons  can  be  pur 
chased  for  $  85.95  ?  Ans.  $  257.85. 

15.  If  15  tons  of  lead  cost  $  105,  what  cost  765  tons  ? 

Ans.  $  5355.00. 

16.  If  16hhd.  of  molasses  cost  $320,  what  cost  176hhd  ? 

Ans.  $  3520.00. 

17.  If  15cwt.  3qr.  171b.  of  sugar  cost  $  124.67,  what  cost 
76cwt.  2qr.  191b.  ?  Ans.  $601.09. 

NOTE.  When  any  of  the  terms  is  a  compound  number,  it  must 
be  reduced  to  the  lowest  denomination  mentioned  in  it ;  therefore,  the 
hundred  weights,  quarters,  &c.,  must  be  reduced  to  pounds,  before 
the  terms  are  multiplied  and  divided  by  each  other. 

18.  If  7s.  6d.  of  the  old  Pennsylvania  currency  are  equal 
to  $  1,  what  is  the  value  of  £76.  19s.  lid.  ? 

Ans.  $205.32|. 

19.  If  8s.  of  the  old  currency  of  New  York  are  equal  to 
$  1,  what  is  the  value  of  £  19.  19s.  8d. 

Ans.  $49.95+. 

20.  If  4s.  8d.  of  the  old  currency  of  South  Carolina  and 
Georgia  are  equal  to  $  1,  what  is  the  value  of  £  1 76.  18s. 
4d.  ?  Ans.  $  758.21+. 

21.  As  4s.  6d.  sterling  of  the  English  currency  are  equal 
to  one  dollar  in  the  United  Sates,  how  many  dollars  are 
there  in  £769.  18s.  9d.  ?  Ans.  $  3421.94+. 


SECT.  42.]  PROPORTION.  141 

22.  If  the  cars  on  the  Boston  and  Portland  Railroad  go 
one  mile  in  2  minutes  and  8  seconds,  how  long  will  they 
be  in  passing  from  Haverhill   to   Boston,  the   distance 
being  32  miles  ?  Ans.  Ih.  8min.  16sec. 

23.  If  one  acre  of  land  cost  $  37.86,  what  cost   144A. 
3R.  17p.  ?  Ans.  $  5484.25+. 

24.  If  a  man  travels  3m.  7fur.  18rd.  in  one  hour,  how 
far  will  he  travel  in  9h.  45min.  19sec.  ? 

Ans.  38m.  2fur.  32-f-rd. 

25.  A  fox  is  96  rods  before  a  greyhound,  and,  while  the 
fox  is  running  15  rods  the  greyhound  will  run  21  rods  ; 
how  far  will  the  dog   run  before  he  can  catch  the  fox  ? 

Ans   336  rods. 

26.  If  5  men  can  reap  a  field  in  12  hours,   how    long 
would  it  take  them  if  4  men  were  added  to  their  num 
ber  ?  Ans.  6f  hours. 

27.  Ten  men  engage  to  build  a  house  in  63  days,  but  3 
of  their  number  being  taken  sick,  how  long  will  it  take 
the  rest  to  complete  the  house  ?  Ans.  90  days. 

28.  If  a  4  cent  loaf  weighs  5  oz.  when  flour  is  $  5   per 
barrel,  what  should  it  weigh  when  flour  is    $  7.50  per 
barrel  ?  Ans.  3£  oz. 

29.  If  7  men  can  mow  a  field  in  ten  days,  when  the  days 
are   14  hours    long,  how  long  would  it  take    the  same 
men  to  mow  the  field,  when  the  days  are  13  hours  long  ? 

Ans.  10|$  days. 

30.  If  291bs.   of  butter  will  purchase  401bs.  of  cheese, 
how  many  pounds  of  butter  will  buy  791bs.  of  cheese  ? 

Ans.  57^1b. 

31.  If  £  of  a  yard  cost  ^  of  a  dollar,  what  will  T£  of  a 
yard  cost  ?  Ans.  8  0.76£. 

STATEMENT.  OPERATION. 

I  :  ft  -  I;  IX  iJ  Xf  =  Jf»=$0.76J  Ans. 

NOTE.     Let  the  pupil  explain,  why  the  first  term  is  inverted  in  the 
operation. 

32.  If  -/T  of  a  gallon  of  oil  cost  T9T  of  a  dollar,  what 
cost  $  of  a  gallon  ?  Ans.  $  1.1 2£. 

STATEMENT.       CANCELLED. 
gal.   gal.    *.    ££    *     Q 

A'*"  A;  ^-X-X  —  =f:=ai.l2£  Ans. 

/'      O     A.  A. 


142  PROPORTION.  [SECT.  42. 

33.  If  4£  yards  of  cloth  cost  $  2£,  what  will  19£  yards 
cost?  Ans.  $11.50. 

STATEMENT.  CANCELLED. 

yd.        yd.  g.         $          grt         OQ 

34.  If  for   4-jZj-  yards  of  velvet,   there   be  received   11£ 
yards  of  calico,  how  many  yards  of  velvet  will  be  suf 
ficient  to  purchase  100  yards  of  calico  ? 

Ans.  39^||  yards. 

35.  If  14$-  ells  English   of  broadcloth  will  pay  for  5T\ 
cwt.  of  sugar,  how  many  yards  will  25I\cwt.  buy  ? 

Ans.  85yd.  3qr.  3f  ana. 

36.  A  certain  piece  of  labor  was  to  have  been  performed 
by  144  men   in  36  days,  but,  a  number  of  them  having 
been   sent   away,  the  work  was  performed  in  48  days  ; 
required  the  number  of  men  discharged. 

Ans.  12  men. 

37  James  can  mow  a  certain  field  in  6  days,  John  can 
mow  it  in  8  days  ;  how  long  will  it  take  John  and  James 
both  to  mow  it  ?  Ans.  3f  days. 

38.  Samuel  can  reap  a  field  of  barley  in  9  hours  ;   but, 
with  the  assistance  of  Alfred,  he  can  reap  it  in  4  hours  ; 
how  long  would  it  take  Alfred  to  reap  it  alone  ?. 

Ans.  7-^  hours. 

39.  A.  Atwood  can  hoe  a  certain  field  in  10  days,  but, 
with  the  assistance  of  his  son  Jerry,  he  can  hoe  it  in  7 
days  ;   and  he  and  his  son  Jacob  can  hoe  it  in  6  days  ; 
how  long  would  it  take  Jerry  and  Jacob  to  hoe  it  to 
gether  ?  Ans.  9^  days. 

40.  Bought  a  horse  for  $  75  ;  for  what  must  I  sell  him 
to  gain  10  per  cent.  ? 

$  100  :  $  110  ::  $  75  :  $  82.50  Ans. 

41.  Bought  40  yards  of  cloth  at  $  5.00  per  yard  ;    for 
what  must  I  sell  the  whole  amount  to  gain  15  per  cent.  ? 

Ans.  $  230.00. 

42.  My  chaise  cost  $  175.00,  but,  having  been  injured, 
I  am  willing  to  sell  it  on  a  loss  of  30  per  cent.  ;  what 
should  I  receive  ?  Ans.  $  122.50. 

43.  Bought  a  cargo  of  flour  on  speculation  at  $  5.00  per 
barrel,  and  sold  it  at  $  6.00  per  barrel  ;   what  did  I  gain 
per  cent.  ?  Ans.  20  per  cent. 


SECT.  43.]  COMPOUND    PROPORTION.  143 

44.  Bought  a  hogshead  of  molasses  for  $  15.00,  but,  it 
not  proving  so  good  as  I  expected,  I  sell  it  for  $  1:2.00; 
what  do  I  lose  per  cent.  ?  Ans.  20  per  cent. 

45.  Sold  a  pair  of  oxen  for  20  per  cent,  less  than  their 
value,  whereas,   I  might  have  sold  them  so  as  to  have 

fained   20   per   cent.,   and,    by  so   doing,    I   have    lost 
60.00  ;  what  was  the  price  for  which  they  were  sold  ? 

Ans.  $  120.00. 

46.  Bought  a  hogshead  of  molasses  for  $  27.50,  at  25 
cents  per  gallon  ;   how  much  did  it  contain  ? 

Ans.  110  gallons. 

47.  A  certain  farm  was  sold  for  $  1728,  it  being  $  15.75 
per  acre  ;   what  was  the  quantity  of  land  ? 

Ans.  109A.  2R.  34fp. 


Section  43. 

COMPOUND    PROPORTION. 

COMPOUND  PROPORTION  is  the  method  of  performing 
by  one  operation,  such  questions  as  require  two  or  more 
operations  in  Single  Proportion. 

1.  If  $  100  will  gain  $  6  in  12  months,  what  will  $  800 
gain  in  8  months  ?  Ans.  $  32.00. 

$  100  :  $  800  I   ..««.«  QO  A 

12  months  :  8  months  }    ' '  $  6  :  $  32  Ans* 

OPERATION. 

800  x  8  x  6 


=  $  32  Ans. 


100  x  12 

The  pupil  will  perceive,  that  the  above  operation  is 
compounded  of  two  statements  in  Single  Proportion, 
which  are  as  follows.  If  $  100  gain  $  6  in  one  year, 
what  will  $  800  gain  in  the  same  time  ?  Ans.  $  48. 


OPERATION. 

$  100  :  $800  ::  $6  :  $48. 


144  COMPOUND    PROPORTION.  [SECT.  43. 

Again,  we  say,    If  $  800  will  gain  $  48  in  12  months, 
what  will  the  same  sum  gain  in  8  months  ?      Ans.  $  32. 

OPERATION. 

12  months  :  8  months  ::  $48  :  $  32  Ans.  as  before. 

This  question  may  be  analyzed  in  the  following  man 
ner.  We  say,  If  $  100  gain  $  6,  $  800  will  gain  8  times 
as  much,  :=  $  48.  Again,  we  say,  If  12  months  gain 
$  48,  1  month  will  gain  -^  of  $  48,  =  $  4,  and,  if  1  month 
gain  $  4,  8  months  will  gain  8  times  $  4,  =  $  32  Answer, 
as  before. 

NOTE.     The  pupil  should  analyze  each  question. 
From  the  above  illustrations,  we  deduce  the  following 
RULE. 

Make  that  number,  which  is  of  the  same  kind  as  the  an 
swer  required,  the  third  term ;  and,  of  the  remaining  num 
bers,  take  any  two,  that  are  of  the  same  kind,  and  consider, 
whether  an  answer,  depending  upon  these  alone,  would  be 
greater  or  less  than  the  third  term,  and  place  them  as  di 
rected  in  Simple  Proportion.  Then  take  any  other  two,  and 
consider,  whether  an  answer,  depending  only  upon  them, 
would  be  greater  or  less  than  the  third  term,  and  arrange 
them  accordingly ;  and  so  on  until  all  are  used.  Multiply 
the  continued  product  of  the  second  terms  by  the  third,  and 
divide  by  the  continued  product  of  the  Jirst,  and  you  pro 
duce  the  answer. 

2.  If  $  100  gain   $  6  in  12  months,  in  how  many  months 
will  8800  gain  $32.  Ans.  8  months. 

3.  If  $100  gain  $6  in  12  months,  how  large  a  sum  will 
it  require  to  gain  $  32  in  8  months  ?  Ans.  $800. 

4.  If  $  800  gain  $  32  in  8  months,  what  is  the  per  cent.  ? 

Ans.  6  per  cent. 

5.  If  15  carpenters  can  build  a  bridge  in  60  days,  when 
the  days  are  15  hours   long,  how  long  will  it  take  20 
men  to  build  the  bridge,  when  the  days  are   10  hours 
long  ?  Ans.  6?£  days. 

6.  If  a  regiment  of  soldiers,  consisting  of  939  men  can 
eat  351  bushels  of  wheat  in  3  weeks,  how  many  soldiers 
will  it  require  to  eat  1404  bushels  in  2  weeks  ? 

Ans.  5634  soldiers. 


SECT.  44.]  COMPANY    BUSINESS.  145 

7.  If  248  men,  in  5£  days  of  11  hours  each,  dig  a  trench 
of  7  degrees  of  hardness,  and  2324-  feet  long,  3§  feet 
wide,  and  2£  feet  deep  ;  in  how  many  days  of  9  hours 
each,  will  24  men  dig  a  trench  of  4  degrees  of  hardness, 
and  337£  feet  long,  5f  feet  wide,  and  3£  feet  deep  ? 

Ans.  132  days. 


Section  44. 

COMPANY    BUSINESS. 

COMPANY  BUSINESS,  or  Fellowship,  is  a  rule,  by  which 
merchants,  and  others  in  partnership,  estimate  their  gain 
or  loss  in  trade.  It  is  of  two  kinds,  single  and  double. 

Single  Fellowship  is,  when  merchants  in  partnership 
employ  their  stock  for  equal  times. 

1.  John  Smith  and  Henry  Grey  enter  into  partnership 
for  three  years,  with  a  capital  of  $  6000,  of  which  Smith 
puts  in  $4000,  and  Grey  $2000.  They  gain  $570. 
What  is  each  man's  share  of  the  gain  ? 

A        (  Smith's  gain  $  380. 

!>  \  Grey's  gain  $190. 

Proof.     $570. 

As  the  whole  stock  is  $  6000,  of  which  $  4000  belongs 
to  Smith,  it  is  evident,  that  his  share  of  the  stock  is 
-£$£§-  =  f  ;  and,  as  each  m£n's  gain  is  in  proportion  to 
his  stock,  f  of  $  570  —  $  380  is  Smith's  share  of  the 
gain.  Grey's  stock  is  $2000,  therefore,  f #$§•  =  £  of 
$  570  =  $  190  is  Grey's  share  of  the  gain. 

Hence,  to  find  any  man's  gain  or  loss  in  trade,  we 
have  the  following 

RULE. 

Multiply  the  whole  gain  or  loss  ly  each  man's  FRACTIONAL 
PART  of  the  stock. 

%.  Three  merchants,  A.,  B.,  and  C.,  engage  in  trade. 
A.  put  in  $  6000,  B.  put  in  $  9000,  and  C.  put  in 

M 


146  DOUBLE    FELLOWSHIP.  [SECT.  45. 

$5000.     They  gain  $840.     What  is  each  man's  share 
of  the  gain  ?  C  A.'s  gain  $  252. 

Ans.  <  B.'s  gain  $  378. 
(  C.'s  gain  $210. 

Proof.     $  840. 

3.  A  bankrupt  owes  Peter  Parker  $8750,  James  Dole 
$  3610,  and  James  Gage   $  7000.     His  effects  sold  at 
auction,  amount  to  $  6875  ;  of  this  sum  $  375  are  to  be 
deducted  for  expenses,  &cc.     What  will  each  receive  of 
the  dividend  ?  C  Parker  $  2937.75|g-£. 

Ans.  \  Dole       $  1212.03T6JT. 
(  Gage      $  2350.20T8/T. 

4.  A  merchant,  failing  in  trade,  owes  A.  $  500,  B.  $  386, 
C.  $  988,  and  D.  $  126.     His  effects  are  sold  for  $  100. 
What  will  each  man  receive  ? 

Ans.   A.   receives    $25.00,   B.    $19.30,   C.    $49.40, 
D.  $  6.30. 


Section    45. 

DOUBLE    FELLOWSHIP. 

When  merchants  in  partnership  employ  their  stock  for 
unequal  times,  it  is  called  Double  Fellowship. 

1.  Josiah  Brown  and  Geqrge  Dole  trade  in  company 
Brown  put  in  $  600  for  8  months,  and  Dole  put  in 
$  400  for  6  months.  They  gain  $  60.  What  is  each 
man's  share  of  the  gain  ? 

Operation  by  analysis.  We  say,  $  600  for  8  months 
is  the  same  as  8  X  $  600  =  $  4800  for  1  month  ;  and 
$  400  for  6  months  is  the  same  as  6  X  $  400  =  $  2400 
for  1  month.  The  question  is,  therefore,  the  same,  as  if 
Brown  had  put  in  $  4800  and  Dole  $  2400  for  1  month 
each.  The  whole  stock  would  then  be  $  4800  +  $  2400 
=  $  7200,  and  Brown's  share  of  the  gain  would  be  ^ftn 
=  §  of  $  60  =  $  40.  Dole's  share  will  be  f|££  =  £~of 
$  60  =  $  20.  Hence  the  propriety  of  the  following 


SECT.  45.]  DOUBLE    FELLOWSHIP.  147 

RULE. 

Multiply  each  marts  stock  by  the  time  it  continued  in 
trade,  and  consider  each  product  a  numerator,  to  be  written 
over  their  sum,  as  a  common  denominator;  then  multiply 
the  whole  gain  or  loss  by  each  fraction,  and  the  several  pro 
ducts  will  be  the  gain  or  loss  of  each  man. 

%.  A.,  B.,  and  C.  trade  in  company.     A.  put  in  $700 
for  5  months  ;  B.  put  in   $  800  for  6  months  ;   and  C 
put  in  $  500  for  10  months.     They  gain   $  399.     What 
is  each  man's  share  of  the  gain  ? 
Ans.  A.'s  gain  $  105,  B.'s  gain  $  144,  C.'s  gain  $  150. 

3.  Leverett  Johnson,  William  Hyde,  and  William  Tyler, 
formed    a    connexion    in    business,    under   the    firm  of 
Johnson,  Hyde,  and  Co.  ;  Johnson  at  first  put  in  8  1000, 
and,  at    the  end  of  6   months,  he  put  in    $  500  more. 
Hyde  at  first  put  in  $  800,  and,  at  the  end  of  4  months, 
he  put  in  $400  more,  but,  at  the  end  of  10  months,  he 
withdrew    $  500  from  the  firm.      Tyler   at   first  put  in 
$  1200,  and,  at  the  end  of  7  months,  he  put  in    $  300 
more,  and,  at  the  end  of  10  months,  he  put  in    $  200. 
At  the  end  of  the  year  they  found  their  net  gain  to  be 
$  1000.     What  is  each  man's  share  ? 

Ans.  Johnson's  gain  $  348.02£ff ,  Hyde's  $273.78^, 
Tyler's  S378.19Tyr 

4.  George  Morse  hired  of  William  Hale,  of  Haverhill, 
his  best  horse  and  chaise  for  a  ride  to  Newburyport,  for 
$  3.00,  with  the  privilege  of  one  person's  having  a  seat 
with  him.     Having  rode  4  miles,  he  took  in  John  Jones 
and  carried  him  to  Newburyport,  and  brought  him  back 
to  the  place  from  which  he  took  him.     What  share  of 
the  expense  should  each  pay,  the  distance  from  Haver- 
hill  to  Newburyport  being  15  miles  ? 

Ans.  Morse  pays  $  1.90,  Jones  pays  $  1.10. 

5.  J.  Jones  and  L.  Cotton  enter  into  partnership  for  one 
year.     January  1,  Jones  put  in  $  1000,  but  Cotton  did 
not  put  in  any   until  the  first  of  April.     WThat  did  he 
then  put  in  to  have  an  equal  share  with  Jones  at  the  end 
of  the  year  ?  Ans.  $  1333.33£. 


148  DUODECIMALS.  [SzcT.46. 

Section   46. 

DUODECIMALS. 

DUODECIMALS  are  so  called  because  they  decrease  by- 
twelves,  from  the  place  of  feet  towards  the  right. 

Inches  are  called  primes,  and  are  marked  thus  '  ;  the 
next  division  after  is  called  seconds,  marked  thus  "  ;  and 
so  on. 

1.  Multiply  8  feet  6  inches  by  3  feet  7  inches. 

As  feet  are  the  integers  of  units,  it 
is  evident,  that  feet  multiplied  by  feet 
will  produce  feet  ;  and,  as  inches  are 
twelfths  of  a  foot,  the  product  of  inches 
by  feet  will  be  twelfths  of  a  foot.  For 

oTj ~i — 7*7r    the  same   reason,  inches  multiplied  by 

inches  will  produce  twelfths  of  an  inch, 
or  one  hundred  and  forty-fourths  of  a  foot.  Hence  we 
deduce  the  following 

RULE. 

Under  the  multiplicand  write  the  same  names  or  denomina 
tions  of  the  multiplier;  that  is,  feet  under  feet,  inches  under 
inches,  SfC.  Multiply  each  term  in  the  multiplicand,  begin 
ning  at  the  lowest,  by  the  feet  of  the  multiplier,  and  write 
each  result  under  its  respective  term,  observing  to  carry  a 
unit  for  every  12  from  each  denomination  to  its  next  supe 
rior.  In  the  same  manner  the  multiplicand  by  the  inches 
of  the  multiplier,  and  write  the  result  of  each  term  one  place 
further  towards  the  right  of  those  in  the  multiplicand. 
Proceed  in  the  same  manner  with  the  seconds,  and  all  the 
rest  of  the  denominations,  and  the  sum  of  all  the  lines  will 
be  the  product  required. 

2.  Multiply  8ft.  3in.  by  7ft.  9in.  Ans.  63ft.  11'  3". 

3.  Multiply  12ft.  9'  by  9ft.  IK  Ans.  126ft.  5'  3". 

4.  Multiply  14ft.  9'  11"  by  6ft.  11'  8". 

Ans.  103ft.  4'  5"  8'"  4"". 


SECT.  46.]  DUODECIMALS.  149 

5.  Multiply  161ft.  8'  6"  by  7ft.  10'.     Ans.  1266ft.  8'  7". 

6.  Multiply  87ft.  1'  11"  by  5ft.  1'  5". 

Ans.  489ft.  8'  0"  2'"  1"". 

7.  What  are  the   contents  of  a  board  18ft.  long  and  1ft. 
lOin.  wide  ?  Ans.  33ft. 

8.  What  are  the  contents  of  a  board  19ft.  Sin.  long  and 
2ft.  11  in.  wide  ?  Ans.  57ft.  4'  4". 

9.  What  are  the  contents  of  a  floor  18ft.  9in.   long  and 
10ft.  6in  wide  ?  Ans.  196ft.  10'  6". 

10.  How  many  square    feet   of  surface    are    there  in  a 
room  14ft.  9in.  long,  12ft.  6in.  wide,  and  7ft.  9in.  high  ? 

Ans.  791ft.  1'  6'. 

11.  John  Carpenter  has  agreed  to  make   12  shoe-boxes 
of  boards  that  are  one  inch  thick.     The  boxes  are  to  be 
3ft.  8in.  long,  1ft.  9in.   wide,   and   1ft.   2in.    high.     How 
many  square  feet  of  boards  will  it  require   to  make  the 
boxes,  and  how  many  cubic  feet  will  they  contain  ? 

Ans.  280  square  feet  ;  66  cubic  feet,  864  inches. 
1£.  My  garden  is  18  rods  long  and  10  rods  wide  ;  a 
ditch  is  dug  round  it  two  feet  wide  and  three  feet  deep, 
but  the  ditch  not  being  of  a  sufficient  breadth  and  depth, 
I  have  caused  it  to  be  dug  one  foot  deeper  and  1ft.  6in. 
wider.  How  many  solid  feet  will  it  require  to  be  re 
moved  ?  Ans.  7540  feet. 

NOTE  1.  A  pile  of  wood,  that  is  8  feet  long,  4  feet  high,  and  4  feet 
wide,  contains  128  cubic  feet,  or  a  cord  ;  and  every  cord  contains  8 
cord-feet;  and,  as  8  is  ^  of  123,  every  cord-foot  contains  16  cubic 
feet;  therefore,  dividing  the  cubic  feet  in  a  pile  of  wood  by  16,  the 
quotient  is  the  cord-feet ;  and,  if  cord-feet  be  divided  by  8,  the  quo 
tient  is  cords. 

When  wood  is  "  corded  "  in  a  pile  4  feet  wide,  by  multiplying  its 
length  by  its  height,  and  dividing  the  product  by  4,  the  quotient  is 
the  cord-feet ;  and,  if  a  load  of  wood  be  8  feet  long,  and  its  height  be 
multiplied  by  its  width,  and  the  product  divided  by  2,  the  quotient  is 
the  cord-feet. 

NOTE  2.     Small  fractions  are  rejected  in  the  operation. 

13.  How  many  cords  of  wood  in  a  pile  56  feet  long,  4 
feet  wide,  and  5  feet  6  inches  high  ?       Ans.  9f  cords. 

14.  How  many  cords  of  wood  in  a  pile  23  feet  8  inches 
long,  4  feet  wide,  and  3  feet  9  inches  high  ? 

Ans.  2T95^  cords 

M* 


150  INVOLUTION.  [SECT.47. 

15.  How  much  wood  in  a  pile  97  feet   long,    3   feet  8 
inches  wide,  and  7  feet  high  ? 

Ans.  19  cords  3f  f  feet. 

16.  If  a  pile  of  wood  be  8  feet  long,   3  feet  9  inches 
wide,  how  high  must  it  be  to  contain  one  cord  ? 

Ans.  4-3*5.  feet. 

17.  If  a  board  be  1  foot  7  inches  wide,  how  long  must  it 
be  to  contain  20  square  feet  ? 

Ans.  12  feet  7^  inches. 

18.  From  a  board  19  feet  7  inches  long,  I  wish  to  slit  off 
one  square  yard  ;   how  far  from  the  edge  must  the  line 
be  drawn  ?  Ans.  5^§^  inches. 

19.  I  have  a  shed  19  feet  8  inches  long,  14  feet  6  inches 
wide,  and  7  feet  6  inches  high  ;  how  many  cords  will  it 
contain  ?  Ans.  16  cords  5|  feet  -)-. 

20.  I  have  a  room   12  feet  long,   11  feet  wide,  and  7£ 
feet  high ;  in  it  are  2  doors,  6  feet  6  inches  high,  and 
30  inches  wide,  and  the  mop-boards  are  8  inches  high  ; 
there  are  3  windows,  3  feet  6  inches  wide,   and  5  feet 
6  inches  high  ;  how  many  square  yards  of  paper  will  it 
require  to  cover  the  walls  ? 

Ans.  25^3-  square  yards. 


Section  47. 

INVOLUTION. 

INVOLUTION  is  the  raising  of  powers  from  any  given 
number,  as  a  root. 

A  power  is  a  quantity  produced  by  multiplying  any 
given  number,  called  a  root,  a  certain  number  of  times 
continually  by  itself;  thus, 

3=    3  is  the  first  power  of     3  — 31. 

3x3=    9  is  the  second  power  of  3  =  32. 

3  X  3  X  3  =  27  is  the  third  power  of   3  =  33. 

3  X  3  X  3  x  3  —  81  is  the  fourth  power  of  3  =  3*. 


SECT.48.]    EVOLUTION.  SQUARE  ROOT.       151 

The  number  denoting  the  power  is  called  the  index,  or 
e-xponent,  of  the  power.  Thus,  the  fifth  power  of  2  is  3*2, 
or  25  ;  the  third  power  of  4  is  64,  or  43. 

To  raise  any  number  to  any  power  required,  we  adopt 
the  following 

RULE. 

Multiply  the  given  number  continually  by  itself,  till  the 
number  of  multiplications  be  one  less,  than  the  index  of  the 
power  to  be  found,  and  the  last  product  will  be  the  power 
required. 

1.  What  is  the  3rd  power  of  5  ?  5  xo  X  5=  125  Ans. 

2.  What  is  the  6th  power  of  4  ?  Ans.  4096. 

3.  What  is  the  4th  power  of  3  ?  Ans.  HI. 

4.  What  is  the  1st  power  of  17  ?  Ans.  17. 

5.  What  is  the  0  power  of  63  ?  Ans.  1. 


Section  48. 

EVOLUTION, 

OR  THE  EXTRACTION  OF  ROOTS. 

EVOLUTION,  or  the  reverse  of  involution,  is  the  extrac 
tion  or  finding  the  roots  of  any  given  power. 

The  root  is  a  number,  whose  continued  multiplication 
into  itself  produces  the  power,  and  is  denominated  the 
square,  cube,  biquadrate,  or  second,  third,  fourth,  &c., 
power,  equal  to  that  power. 

Thus,  4  is  the  square  root  of  16,  because,  4x4  =  16; 
and  3  is  the  cube  root  of  27,  because,  3  X  3  X  3  =  27; 
and  so  on. 

Roots,  which  approximate,  are  surd  roots ;  and  those, 
which  are  perfectly  accurate,  are  called  rational  roots. 

EXTRACTION  OF  THE  SQUARE  ROOT. 
1.  What  is  the  square  root  of  625  ? 

To  illustrate  this  question,  we  will  suppose,  that  we 


152  SQUARE    ROOT.  [SECT.  48. 

have  625  tile,  each  of  which  is  one  foot  square  ;  we  wish 
to  know  the  side  of  a  square  room,  whose  floor  they  will 
pave  or  cover.  If  we  find  a  number  multiplied  into  itself, 
that  will  produce  625,  that  number  will  be  the  side  of  a 
square  room,  which  will  require  625  tiles  to  cover  its 
floor.  We  perceive  that  our  number  (625)  consists  of 
three  figures,  therefore,  there  will  be  two  figures  in  the 
root  ;  for  the  product  of  any  two  numbers  can  have,  at 
most,  but  just  so  many  figures,  as  there  are  in  both  fac 
tors,  and,  at  least,  but  one  less.  We  will,  therefore,  for 
convenience,  divide  our  number  (625)  into  two  parts, 

called  periods,  writing    a   point 
OPERATION.  over  the  right  hand    figure   of 

625(25Ans.        each  period;    thus,   625.     We 

now  find,  that  the  greatest  square 

45)225  number  in  the  left  hand  period, 

225  6  (hundred),  is  4    (hundred)  ; 

and  that  its  root  is  2,  which  we 

write  in  the  quotient  (see  operation).  As  this  2  is  in 
the  place  of  tens,  its  value  must  be  20  and  its  square 
400. 

Let  this  be  represented  by  a  square,  FIG.  i. 

whose  sides  measure  20  feet  each,  and  20 

whose  contents  will,  therefore,  be  400 
square  feet.  (See  figure  1.)  We  now 
subtract  400  from  625,  and  there  re 
mains  225  square  feet,  to  be  arranged 
on  two  sides  of  figure  1,  in  order  that 
its  form  may  remain  square.  We 
therefore  double  the  root  20,  one  of  20 

the  sides,  and  it  gives  the  length  of 
the  two  sides  to  be  enlarged  ;   viz.  40.     We  then  inquire, 
how  many  times  40,  as  a  divisor,  is  contained  in  the   divi 
dend,  and  find  it  to  be  5  times  ;  this  we  write  in  the  root, 
and  also  in  the  divisor. 

This  5  is  the  breadth  of  the  addition  to  our  square. 
(See  figure  2.)  And  this  breadth,  multiplied  by  the 
length  of  the  two  additions  (40)  gives  the  contents  of  the 
two  figures,  E  and  F,  200  square  feet,  which  is  100 
feet  for  each. 

There  now  remains  the  figure  G,  to  complete  the 
square,  each  side  of  which  is  5  feet  ;  it  being  equal  to 


20 


D 

20 
20 

400 


20 


SECT.  48.] 


SQUARE    ROOT. 


153 


FIG.  II. 

20 

E20 
5 

G  5 
5 

100 

25 

D20 
20 

20 
5 

400 

100 
F 

'20 


D  contains  400  square  feet 


E  do. 
F  do. 
G  do. 

Proof. 


100 

100 

25 

625 


dc 

do. 

do. 


do. 
do. 
do. 


the  breadth  of  the  additions  E 
and  F.  Therefore,  if  we  square 
5,  we  have  the  contents  of  the 
last  addition,  G  =  25.  It  is  on 
account  of  this  last  addition,  that 
the  last  figure  of  the  root  is  placed 
in  the  divisor.  If  we  now  multiply 

the  divisor,  45,  by  the  last  figure  2Q         ~-'  " "     20 

in  the  root  (5),  the  product  will 
be  '225,  which  is  equal  to  the  re 
maining  feet,  after  we  have  form 
ed  our  first  square,  and  equal  to 
the  additions  E,  F,  and  G,  in  fig 
ure  2.  We  therefore  perceive, 
that  figure  2  may  represent  a 
floor  25  feet  square,  containing 
625  square  feet.  From  the  above, 
we  infer  the  following  25X25  =  625. 

RULE. 

1.  Distinguish   the  given  number  into   periods  of  two 
figures  each,  by  putting  a  point  over  the  place  of  units,  an 
other  over  the  place  of  hundreds,  and  so  on,  which  points 
show  the  number  of  Jigures  the  root  will  consist  of. 

2.  Find  the  greatest   square  number  in  the  first  or  left 
hand  period,  place  the  root  of  it  at  the  right  hand  of  the 
given  number,  (after  the  manner  of  a  quotient  in  division,) 
for  the  first  figure  of  the  root,  and  the  square  number  under 
the  period,  and  subtract,  it  therefrom,  and  to  the  remainder 
bring  down  the  next  period  for  a  dividend. 

3.  Place   the  double  of  the  root  already  found,  on  the 
left  hand  of  the  dividend  for  a  divisor. 

4.  Seek  how  often  the  divisor  is  contained  in  the  divi 
dend,  (except  the  right  hand  figure,)  and  place  the  answer 
in  the  root  for  the  second  figure  of  it,,  and  likewise  on  the 
right  hand  of  the  divisor.     Multiply  the  divisor  with  the 
figure  last  annexed  by  the  figure  last  placed  in  the  root,  and 
subtract  the  product  from  the  dividend.      To  the  remainder 
join  the  next  period  for  a  new  dividend. 

5.  Double  the  figures  already  found  in  the  root  for  a  new 


154  SQUARE    ROOT.  [SECT.  48. 

divisor,  (or,  bring  down  your  last  divisor  for  a  new  one, 
doubling  the  right  hand  figure  of  it,}  and  from  these  find 
the  next,  figure  in  the  root,  as  last  directed,  and  continue 
the  operation  in  the  same  manner,  till  you  have  brought 
down  all  the  periods. 

NOTE  1  .  If,  when  the  given  power  is  pointed  off,  as  the  power 
requires,  the  left  hand  period  should  be  deficient,  it  must  nevertheless 
stand  as  the  first  period. 

NOTE  2.  If  there  be  decimals  in  the  given  number,  it  must  be 
pointed  both  ways  from  the  place  of  units.  If,  when  there  are  inte 
gers,  the  first  period  in  the  decimals  be  deficient,  it  may  be  completed 
by  annexing  so  many  ciphers  as  the  power  requires.  And  the  root 
must  be  made  to  consist  of  so  many  whole  numbers  and  decimals,  as 
there  are  periods  belonging  to  each  ;  and  when  the  periods  belonging 
to  the  given  numbers  are  exhausted,  the  operation  may  be  continued 
at  pleasure  by  annexing  ciphers. 

NOTE  3.  If  it  be  required  to  extract  the  square  root  of  a  vulgar 
fraction,  reduce  the  fraction  to  its  lowest  terms,  then  extract  the 
square  root  of  the  numerator  for  a  new  numerator,  and  of  the  denomi 
nator  for  a  new  denominator;  or,  reduce  the  vulgar  fraction  to  a 
decimal,  and  extract  its  root. 


What  is  the  square  root  of  148996 


? 

OPERATION 

148990(386 
j) 

68)589 
544 

766)4596 
4596 

3.  What  is  the  square  root  of  23804641  ?  Ans.  4879. 

4.  What  is  the  square  root  of  10673289  ?  Ans.  3267. 

5.  What  is  the  square  root  of  20894041  ?  Ans.  4571. 

6.  What  is  the  square  root  of  1014049  ?  Ans.  1007. 

7.  What  is  the  square  root  of  516961  ?  Ans.  719. 

8.  What  is  the  square  root  of  182329  ?  Ans.  427. 

9.  What  is  the  square  root  of  61723020.96  ? 

Ans.  7856.4. 

10.  What  is  the  square  root  of  9754.60423716  ? 

Ans.  98.7654. 

11.  What  is  the  square  root  of  fff£  ?  Ans>  Sr- 


SECT.  48.]  SQUARE    ROOT.  155 


12.  What  is  the  square  root  of  TyVW  ?  Ans 

13.  What  is  the  square  root  of  -^  ?  Ans.  j^. 

14.  What  is  the  square  root  of  £ff  ?  Ans.  £f. 

15.  What  is  the  square  root  of  60T^  ?  Ans.  7£. 

16.  What  is  the  square  root  of  28-|£  ?  Ans.  .5| 

17.  What  is  the  square  root  of  47|J-  ?  Ans.  6J. 

APPLICATION  OF  THE  SQUARE  ROOT. 

18.  A  certain  general  has  an  army  of  226576  men  ;  how 
many  must  he  place  rank  and  file  to  form  them  into  a 
square  ?  Ans.  4V6. 

NOTE.     In  a  right  angle  triangle,  the  square  of  the  longest  side  is 
equal  to  the  sum  of  the  squares  of  the  other  two  sides. 

19.  What  must  be  the  length  of  a  ladder  to  reach  to  the 
top  of  a  house  40  feet  in  height  ;  the  bottom  of  the  lad 
der  being  placed  9  feet  from  the  sill  ?        Ans.  41  feet. 

20.  Two  vessels  sail  from  the  same  port ;  one  sails  due 
north  360   miles,    and   the    other   due  east  450   miles  ; 
what  is  their  distance  from  each  other  ? 

Ans.  576.2-f-  miles. 

21.  If  a  pipe,  2  inches  in  diameter,  will  fill  a  cistern  in 
20£  minutes,   how  long   would  it  take  a  pipe,  that  is  3 
inches  in  diameter  ?  Ans.  9  minutes. 

22.  If  an    anchor,  which  weighs  2000  Ibs.,  requires  a 
cable  3  inches  in  diameter,  what  should  be  the  diameter 
of  a  cable,  when  the  anchor  weighs  40001bs.  ? 

Ans.  4.24-j-  inches. 

23.  How    large    a    square    stick    may  be  hewn   from  a 
round  one,  which  is  30  inches  in  diameter  ? 

Ans.  21.2-J-  inches  square. 

24.  John  Snow's  dwelling  is  60  rods  north  of  the  meet 
inghouse,  James  Briggs'  is  80  rods  east  of  the  meet 
inghouse,    Samuel  Jenkins'  is  70  rods  south,  and  James 
Emerson's  90  rods  west  of  the  meetinghouse  ;  how  far 
will   Snow  have  to  travel  to  visit  his  three  neighbours, 
and  then  return  home  ?  Ans.  428. 4-|~  rods. 


156  CUBE    ROOT.  [SECT. 49. 

Section  49. 

EXTRACTION   OF  THE   CUBE  ROOT. 

A  CUBE  is  a  solid,  bounded  by  six  equal  squares. 

A  number  is  said  to  be  cubed,  when  it  is  multiplied 
into  its  square. 

To  extract  the  cube  root,  is  to  find  a  number,  which, 
being  multiplied  into  its  square,  will  produce  the  given 
number. 

The  extraction  of  this  root  has  been  illustrated  by 
mathematicians  in  various  ways.  But  it  is  believed,  that 
Robert  Record,  Esquire,  of  London,  in  his  Arithmetic 
published  in  1673,  was  among  the  first,  who  illustrated 
this  rule  by  the  use  of  various  diagrams  and  blocks. 
The  same  thing,  with  but  little  variation,  has  been  done 
by  several  arithmeticians  in  our  own  country. 

The  Rule  for  extracting  the  root  depends  on  the  fol 
lowing 

THEOREM. 

If  any  line  or  number  be  divided  into  two  parts,  the 
cube  of  the  whole  line  or  number,  is  equal  to  the  cube 
of  the  greater  part,  plus  the  square  of  the  greater  part 
multiplied  by  3  times  the  less  part,  plus  the  square  of 
the  less  part  multiplied  by  3  times  the  larger  part,  plus 
the  cube  of  the  less  part. 

To  illustrate  this  Theorem,  let  27  be  divided  into  two 
parts,  20  and  7.  Then,  by  the  hypothesis,  the  cube  of 
27  is  equal  to  the  cube  of  20,  plus  the  square  of  20  mul 
tiplied  by  3  times  7,  plus  the  square  of  7  multiplied  by 
3  times  20,  plus  the  cube  of  7. 

OPERATION. 

Cube  of  27  =  19683 

Cube  of  20  =8000 

Square  of  '20  multiplied  by  3  times  7  =     8400 

Square  of  7  multiplied  by  3  times  20  =     2940 

Cube  of  7  =        343 

Proof.  =  1  9  68  3~ 


SECT.  49.]  CUBE   ROOT.  157 

Hence  the  following 

RULE. 

1.  Separate  the  given  number  into  periods  of  three  figures 
each)  by  putting  a  point  over  the  unit  figure,  and  every  third 
figure  beyond  the  place  of  units. 

2.  Find  by  the  table  the  greatest  cube  in  the  left  hand 
period,  and  put  its  root  in  the  quotient. 

3.  Subtract  the  cube,  thus  found,  from  this  period,  and 
to  the  remainder  bring  down  the  next  period ;  call  this  the 
dividend. 

4.  Multiply  the  square  of  the  quotient  by  300,  calling  it 
the  triple  square ;  multiply  also  the  quotient  by  30,  calling 
it  the  triple  quotient ;  the  sum  of  these  call  the  divisor. 

5.  Find  how  many  times  the  divisor  is  contained  in  the 
dividend,  and  place  the  result  in  the  quotient. 

6.  Multiply  the  triple  square  by  the  last  quotient  figure, 
and  write  the  product  under  the    dividend ;   multiply  the 
square  of  the  last  quotient  figure  by  the  triple  quotient,  and 
place  this  product  under  the  last ;  under  all,  set  the  cube  of 
the  last  quotient  figure,  and  call  their  sum  the  subtrahend. 

7.  Subtract  the  subtrahend  from  the  dividend,  and  to  the 
remainder  bring  down  the  next  period  for  a  new  dividend, 
with  which  proceed  as  before,  and  so  on,  till  the  whole  is 
completed. 

NOTE  1 .  The  same  rule  must  be  observed  for  continuing  the  oper 
ation,  and  pointing  for  decimals,  as  in  the  square  root. 

NOTE  2.  In  inquiring  how  many  times  the  dividend  will  contain 
the  divisor,  we  must  sometimes  make  an  allowance  of  two  or  three 
units.  See  National  Arithmetic,  page  205. 

1.  What  is  the  cube  root  of  78402752  ? 

OPERATION. 

78402752(428  4x4x300=       4800 

64  4x30=         120 

4920) H402=lst  dividend.  1st  divisor.=  4920 

9600  4800x2=  9600 

480  120x2x2=  430 

8  2x2x2=  8 

10088=lst  subtrahend.  1st  subtrahend. =  10088 


158  APPLICATION  OF  THE    CUBE  ROOT.   [SECT.  49. 

530460)4314752— 2d  dividend.        42x42x300=  529200 

4233600  42x30=       1260 

80640  2d  divisor.=  530460 

512  529200  x8=4233600 

43ll752=2d  subtrahend.      1260x8x8=     80640 

8x8x8=        512 
2d  subtrahend. =4314752 

2.  What  is  the  cube  root  of  74088  ?  Ans.  42. 

3.  What  is  the  cube  root  of  185193  ?  Ans.  57. 

4.  What  is  the  cube  root  of  80621568  ?  Ans.  432. 

5.  What  is  the  cube  root  of  176558481  ?  Ans.  561. 

6.  What  is  the  cube  root  of  257259456  ?  Ans.  636. 

7.  What  is  the  cube  root  of  1860867  ?  Ans.  123. 

8.  What  is  the  cube  root  of  1879080904  ?  Ans.  1234. 

9.  What  is  the  cube  root  of  41673648.563  ? 

Ans.  346.7. 

10.  What  is  the  cube  root  of  48392.1516051  ? 

Ans.  78.51. 

11.  What  is  the  cube  root  of  8.144865728  ? 

Ans.  2.012. 

12.  What  is  the  cube  root  of  £^  ?  Ans.  &• 

13.  What  is  the  cube  root  of  49/T  ?  Ans.  3f . 

14.  What  is  the  cube  root  of  166f  ?  Ans.  5J. 

15.  What  is  the  cube  root  of  85-$^  ?  Ans.  4f . 

APPLICATION  OF  THE  CUBE  ROOT. 

Spheres  are  to  each  other,  as  the  cubes  of  their  diam 
eter. 

Cones  are  to  each  other,  as  the  cubes  of  their  altitudes 
or  bases. 

All  similar  solids  are  to  each  other,  as  the  cubes  of 
their  homologous  sides. 

16.  If  a  ball,  4  inches  in  diameter,  weighs  SOlbs.,  what 
is  the  weight  of  a  ball  6  inches  in  diameter  ? 

Ans.  168.7+  Ibs. 

17.  If  a  sugar  loaf,  which  is  12  inches  in  height,  weighs 
161bs.,  how  many  inches  may  be  broken  from  the  base, 
that  the  residue  may  weigh  Slbs.  ?  Ans.  2.5-}-  in. 


SECT.  50.]  GEOMETRICAL    PROBLEMS.  159 

18.  If  an  ox,  that  weighs  SOOlbs.,  girts  6  feet,  what  is  the 
weight  of  an  ox  that  girts  7  feet  ?  Ans.  1270. 31bs. 

19.  If  a  tree,  that    is  one  foot  in  diameter,  make  one 
cord,  how  many  cords  are  there  in  a  similar  tree,  whose 
diameter  is  two  feet  ?  Ans.  8  cords. 

20.  If  a  bell,  30  inches  high,  weighs  lOOOlbs.,  what  is  the 
weight  of  a  bell  40  inches  high  ?  Ans.  2370. 31bs. 

21.  If  an   apple,  6  inches  in  circumference,  weighs  16 
ounces,  what  is  the  weight  of  an  apple  12  inches  in  cir 
cumference  ?  Ans.  128  ounces. 


Section  5O. 

GEOMETRICAL    PROBLEMS. 

1.  To  find  the  area  of  a  square  or  parallelogram. 

RULE.  Multiply  the  length  by  the  breadth,  and  the  pro 
duct  is  the  superficial  contents. 

2.  To  find  the  area  of  a  rhombus  or  rhomboid. 

RULE.  Multiply  the  length  of  the  base  by  the  perpen 
dicular  height. 

3.  To  find  the  area  of  a  triangle. 

RULE.  Multiply  the  base  by  half  the  perpendicular 
height ;  or,  add  the  three  sides  together  ;  then  take  half  of 
that  sum,  and  out  of  it  subtract  each  side  severally  ;  multiply 
the  half  of  the  sum  and  these  remainders  together,  and  the 
square  root  of  this  product  will  be  the  area  of  the  tri 
angle. 

4.  Having  the  diameter  of   a   circle  given,  to  find   the 
circumference. 

RULE.  Multiply  the  diameter  by  3.141592,  and  the  pro 
duct  is  the  circumference. 


160  GEOMETRICAL    PROBLEMS.  [SECT.  60. 

NOTE.  The  exact  proportion,  which  the  diameter  of  a  circle  bears 
to  the  circumference,  has  never  been  discovered,  although  some 
mathematicians,  have  carried  it  to  200  places  of  decimals  If  the 
diameter  of  a  circle  be  1  inch,  the  circumference  will  be  3.141592653 
5897932384626433832795028841971693993751058209749445923078164062 
8620899862S03482534211706798214808651328230664709384464609550518 
22317253594081284802  inches  nearly. 

5.  Having  the  diameter  of  a  circle  given,  to  find  the  side 
of  an  equal  square. 

RULE.  Multiply  the  diameter  by  .886227,  and  the  pro 
duct  is  the  side  of  an  equal  square. 

6.  Having  the  diameter  of  a  circle  given,  to  find  the  side 
of  an  equilateral  triangle  inscribed. 

RULE.  Multiply  the  diameter  by  .707016,  and  the  pro 
duct  is  the  side  of  a  triangle  inscribed. 

7.  Having  the  diameter  of  a  circle  given,  to  find  the  area. 

RULE.  Multiply  the  square  of  the  diameter  by  .785398, 
and  the  product  is  the  area.  Or,  multiply  half  the  diam 
eter  by  half  the  circumference,  and  the  product  is  the  area. 

8.  Having  the  circumference  of  a  circle  given,  to  find  the 
diameter. 

RULE.  Multiply  the  circumference  by  .31831,  and  the 
product  is  the  diameter. 

9.  Having  the  circumference  of  a  circle  given,  to  find 
the  side  of  an  equal  square. 

RULE.  Multiply  the  circumference  by  .282094,  and  the 
product  is  the  side  of  an  equal  square. 

10.  Having  the  circumference  of  a  circle  given,  to  find 
the  side  of  an  equilateral  triangle  inscribed. 

RULE.  Multiply  the  circumference  by  .2756646,  and 
the  product  is  the  side  of  an  equilateral  triangle  inscribed. 

11.  Having  the  circumference  of  a  circle  given,  to  find 
the  side  of  an  inscribed  square. 

RULE.  Multiply  the  circumference  by  .225079,  and  the 
product  is  the  side  of  a  square  inscribed. 


SECT.  60.]  GEOMETRICAL    PROBLEMS.  101 

12.  To  find  the  contents  of  a  cube  or  parailelopipedon. 

RULE.  Multiply  the  length,  height,  and  breadth,  con 
tinually  together,  and  the  product  is  the  contents. 

13.  To  find  the  solidity  of  a  prism. 

RULE.  Multiply  the  area  of  the  base,  or  end,  by  the 
height. 

14.  To  find  the  solidity  of  a  cone  or  pyramid. 

RULE.     Multiply  the  area  of  the  base  by  -J-  of  its  height. 

15.  To  find  the  surface  of  a  cone. 

RULE.  Multiply  the  circumference  of  the  base  by  half 
its  slant  height. 

16.  To  find  the  solidity  of   the   frustum  of  a  cone,  or 
pyramid. 

RULE.  Multiply  the  diameters  of  the  two  bases  together, 
and  to  the  product  add  ^  of  the  square  of  the  difference  of 
the  diameters ;  then  multiply  this  sum  by  .785398,  and  the 
product  will  be  the  mean  area  between  the  two  bases ;  lastly, 
multiply  the  mean  area  by  the  length  of  the  frustum,  and 
the  product  will  be  the  solid  contents. 

Or,  find  when  it  would  terminate  in  a  cone,  and  then 
find  the  contents  of  the  part  supposed  to  be  added,  and  take 
it  away  from  the  whole. 

17.  To  find  the  solidity  of  a  sphere  or  globe. 
RULE.     Multiply  the  cube  of  the  diameter  by  .5236. 

18.  To  find  the  convex  surface  of  a  sphere  or  globe. 
RULE.     Multiply  its  diameter  by  its  circumference. 

19.  To  find  the  contents  of  a  spherical  segment. 

RULE.  From  three  times  the  diameter  of  the  sphere, 
take  double  the  height  of  the  segment ;  then  multiply  the  re 
mainder  by  the  square  of  the  height,  and  the  product  by  the 
decimal  .5236  for  the  contents  ;  or  to  three  times  the  square 
of  the  radius  of  the  segment's  base,  add  the  square  of  its 

N* 


162  GEOMETRICAL   PROBLEMS.  [SECT.  50. 

height ;  then  multiply  the  sum  by  the  height,  and  the  product 
by  .5236  for  the  contents. 

HO.  To  find   how  large  a  cube   may  be    cut    from   any 
given  sphere,  or  be  inscribed  in  it. 

RULE.  Square  the  diameter  of  the  sphere,  divide  that 
product  by  3,  and  extract  the  square  root  of  the  quotient 
for  the  answer. 

SI.  To   find   the   number  of  gallons,  &c.,  in  a   square 
vessel. 

RULE.  Take  the  dimensions  in  inches  ;  then  multiply  the 
length,  breadth,  and  height,  together  ;  divide  the  product  by 
282  for  ale  gallons,  231  for  wine  gallons,  and  2150.42  for 
bushels. 

%%.  To  find  the  contents  of  a  cask. 

RULE.  Take  the  dimensions  of  the  cask  in  inches  ;  viz. 
the  diameter  of  the  bung  and  head,  and  the  length  of  the 
cask.  Note  the  difference  between  the  bung  diameter  and 
the  head  diameter.  If  the  staves  of  the  cask  be  much  curved 
between  the  bung  and  the  head,  multiply  the  difference  by  .7; 
if  not.  quite  so  much  curved,  by  .65 ;  if  they  bulge  yet  less, 
by  .6 ;  and,  if  they  are  almost  straight,  by  .55  ;  add  the 
product  to  the  head  diameter ;  the  sum  will  be  a  mean  diam 
eter  by  which  the  cask  is  reduced  to  a  cylinder. 

Square  the  mean  diameter  thus  found,  then  multiply  it  by 
the  length  ;  divide  the  product  by  359  for  ale  or  beer  gal 
lons,  and  by  294  for  wine  gallons. 

23.  To  find  the  contents  of  a  round  vessel,  wider  at  one 
end  than  the  other. 

RULE.  Multiply  the  greater  diameter  by  the  less;  to 
this  product,  add  ^  of  the  square  of  their  difference,  then 
multiply  by  the  height,  and  divide  as  in  the  last  rule. 

24.  To  measure  round  timber. 

RULE.  Multiply  the  length  of  the  stick,  taken  in  feet,  by 
the  square  of  £  the  girt,  taken  in  inches ;  divide  this  pro 
duct  by  144,  and  the  quotient  is  the  contents  in  cubic  feet. 


SECT.  50.]  GEOMETRICAL    PROBLEMS.  163 

NOTE.     The  girt  is  usually  taken  about  *  the  distance  from  the 
larger  to  the  smaller  end. 

1.  What  are  the  contents  of  a  board  25  feet  long  and  3 
feet  wide  ?  Ans.  75  feet. 

2.  What  is  the  difference  between  the  contents  of  two 
floors  ;   one  is  37  feet  long  and  27  feet  wide,  and  the 
other  is  40  feet  long  and  20  feet  wide  ?     Ans.  199  feet. 

3.  The  base  of  a  rhombus  is  15  feet,  and  its  perpendicu 
lar  height  is  12  feet  ;  what  are  its  contents  ? 

Ans.  180  feet. 

4.  What  are  the  contents  of  a  triangle,  whose  base  is  24 
feet,  and  whose  perpendicular  height  is  IS  feet  ? 

Ans.  216  feet. 

5.  What  are  the  contents  of  a  triangular  piece  of  land, 
whose  sides  are  50  rods,  60  rods,  and  70  rods  ? 

Ans.  1469.69+  rods. 

6.  What  is  the  circumference  of  a  circle,  whose  diame 
ter  is  50  feet  ?  Ans.  157.0796+  feet. 

7.  We  have  a  round  field  40  rods  in  diameter  ;   what  is 
the  side  of  a  square  field,   that  will  contain  the  same 
quantity  ?  Ans.  35.44+  rods. 

8.  What  is  the  side  of  an  equilateral  triangle,  that  may 
be  inscribed  in  a  circle  50  feet  in  diameter  ? 

Ans.  35.35+  feet. 

9.  If  the  diameter  of  a  circle  be  200  feet,  what  is  the 
area  ?  Ans.  31415.92+  feet. 

10.  What  is  the    diameter  of  a  circle,  whose  circumfer 
ence  is  80  miles  ?  Ans.  25.46+  miles. 

11.  I  have  a  circular  field  100  rods  in  circumference  ; 
what  must  be  the  side  of  a  square  field,  that  shall  con 
tain  the  same  area  ?  Ans.  28.2+  rods. 

12.  Required  the  side    of  a   triangle,   that  may  be   in 
scribed  in  a  circle,  whose  circumference  is  1000  feet. 

Ans.  275.66+  feet. 

13.  How  large  a  square  field  may  be  inscribed  in  a  cir 
cle,  whose  circumference  is  100  rods  ? 

Ans.  22.5+  rods  square. 

14.  How  many  cubic   feet   are  there   in  a  cube  whose 
sides  are  8  feet  ?  Ans.  512  feet. 

15.  What  is  the  difference  between  the  number  of  cubic 
feet  in  a  room  30  feet  long,  20  feet  wide,   and   10  feet 


164  GEOMETRICAL    PROBLEMS.  [SECT.  60. 

high,  and  the  number  of  square  feet  in  the  surface  of  the 
room  ?  Ans  (5000  solid  feet.     2200  square  feet. 

16.  What  are  the  contents  of  a  triangular  prism,  whose 
length  is  20  feet,  and  the  three  sides  of  its  triangular 
end  or  base  5,  4,  and  3  feet  ?  Ans.  120  feet. 

17.  What  are  the  solid  contents  of  a  cone,  whose  height 
is  30  feet,  and  the  diameter  of  its  base  5  feet  ? 

Ans.  196.3+  feet. 

18.  The  largest  of  the  Egyptian  pyramids  is  square  at 
its  base,  and  measures  693  feet  on  a  side.     Its  height 
is  500  feet.     Now,  supposing  it  to  come  to  a  point  at  its 
vertex,  what  are  its  solid  contents,  and  how  many  miles 
in  length  of  wall  would  it  make,  4  feet  in  height  and  2 
feet  thick  ? 

Ans.  80,041,500  cubic  feet.     1894.9  miles  in  length. 

19.  Required  the  convex  surface  of  a  cone,  whose  side 
is  50  feet,  and  the  cirqumference  at  its  base  12  feet. 

Ans.  300  feet 

20.  Required  the  solid  contents  of  Bunker  Hill  monu 
ment,  whose  height  is  220  feet,  and  being  30  feet  square 
at  its  base,  and  15  feet  square  at  its  vertex. 

Ans.  115500  cubic  feet. 

£1.  What  are  the  contents  of  a  stick  of  timber  20  feet 
long,  and  the  diameter  at  the  larger  end  12  inches,  and 
at  the  smaller  end  6  inches  ?  Ans.  9.163-j-  feet. 

22.  What  is  the  solidity  of  a  sphere,  whose  diameter  is 
20  inches  ?  Ans.  4183.8+  inches. 

23.  What  is  the  convex  surface  of  a  globe,  whose  diam 
eter  is  20  inches  ?  Ans.  1256.6+  inches. 

24.  What  are  the  contents  of  a  spherical   segment  3  feet 
in  height,  cut  from  a  sphere  10  feet  in  diameter  ? 

Ans.  113.0976  feet. 

25.  What  is  the  solidity  of  a  segment  of  a  sphere,  its 
height  being  8  inches,  and  the  diameter  of  its  base  20 
inches  ?  Ans.  1224.7232  inches. 

2G.  How  large  a  cube  may  be  inscribed  in  a  sphere  10 
inches  in  diameter  ?  Ans.  5.773+  inches. 

27.  How  many  wine  gallons  will  a  cubical  box  contain, 
that  is  8  feet  long,  4  feet  high,  and  3  feet  wide  ? 

Ans.  718.1+  gallons. 

28.  How  many  bushels  of  grain  will  a  box  contain,  that 
is  12  feet  long,  5  feet  wide,  and  4  feet  high  ? 

Ans.  192.8+  bushels. 


SECT. 51.]         MISCELLANEOUS   QUESTIONS.  165 

29.  What  are  the  contents  of  a  cask,  in  wine  gallons, 
whose   bung  diameter  is  30   inches,    head    diameter  24 
inches,  and  length  40  inches  ?     Ans.  108.19-f-  gallons. 

30.  How  many  cubic  feet  in  a  stick  of  timber,  which  is 
40  feet  long,  and  whose  girt  is  GO  inches  ? 

Ans.  62£  feet. 


Section  51. 

MISCELLANEOUS    QUESTIONS. 

1.  What  is  the  difference  between  7  pence  and  10  cents? 

Ans.  id. 

2.  What  number  is  that,   to   which,   if  •§•  be  added,  the 
sum  will  be  7^  ?  Ans.  7f. 

3.  What   number  is  that,  from  which,  if  3y  be  taken,  the 
remainder  will  be  4|-  ?  Ans.  7£f. 

4.  What  number  is  that,  to  which,  if  3^  be  added,  and 
the  sum  divided  by  5£,  the  quotient  will  be  5  ? 

Ans.  23f 

5.  From  T7T  of  a  mile  take  J  of  a  furlong. 

Ans.  4fur.  12rd.  8ft.  Sin. 

6.  From  7  acres  take  T9T  of  a  rood. 

Ans.  6A.  3R.  7p.  74ft.  36in. 

7.  John   Swift  can  travel  7  miles  in  f  of  an  hour,   but 
Thomas  Slow  can  travel  only  5  miles  in  ^  of  an  hour. 
Both  started  from  Danvers  at  the  same  time   for  Boston, 
the  distance    being  12  miles.      How  much    sooner  will 
Swift  arrive  in  Boston  than  Slow  ?  Ans.  12£|-  seconds. 

9.  If  f  of  a  ton  cost  849,  what  cost  Icwt.  ? 

Ans.  $  3.92. 

9.  How  many  bricks,  8  inches  long,  4  inches  wide,  and 
2  inches  thick,  will   it  take  to  build  a  wall  40  feet  long, 
20  feet  high,  and  2  feet  thick  ?  Ans.  43200  bricks. 

10.  How  many  bricks  will  it  take  to  build  the  walls  of  a 
house,  which  is  80  feet  long,   40  feet  wide,  and  25  feet 
high,  the  wall  to  be  12  inches  thick  ;  the  brick  being  of 
the  same  dimensions,  as  in  the  last  question  ? 

Ans.  159300  bricks. 


166  MISCELLANEOUS    QUESTIONS.         [SECT.  51. 

11.  How  many  tiles,  8  inches  square,  will  cover  a  floor 
18  feet  long,  and  12  feet  wide  ?  Ans.  486  tiles. 

12.  If  it  cost  $  18.25  to  carry  llcwt.  3qr.  191bs.  40  miles, 
how  much  must  be  paid  for  carrying  83cwt.  2qr.   lllbs. 
96  miles  ?  Ans.  $  267. 1224ffW- 

13.  A  merchant  sold  a  piece  of  cloth  for  $24,  and  there 
by  lost  25  per  cent.  ;  what  would  he  have  gained,  had 
he  sold  it  for  $  34  ?  Ans.  6£  per  cent. 

14.  Bought  a  hogshead  of  molasses,  containing  120  gal 
lons,  for    $30;    but  20  gallons  having  leaked  out,  for 
what  must  I  sell  the  remainder  per  gallon  to  gain   $  10  ? 

Ans.  $  0.40. 

15.  In  a  piece  of  land  117^  rods  long,  and  11 2f  rods 
wide,  how  many  acres  ? 

Ans.  82A.  1R.  18p.  2yd.  7ft.  133^in. 

16.  Bought  a  quantity  of  goods  for  $  128.25,  and,  having 
kept  them  on  hand  6  months,  for  what  must  I  sell  them 
to  gain  6  per  cent.  ?  Ans.  $  140.02. 

17.  If  27  bushels  of  potatoes  cost  $8.75,  what  must  be 
paid  for  36  bushels  ?  Ans.  $  11.60-J-. 

18.  How  many  bushels  of  oats,  at  50  cents  per  bushel, 
must  I  give  Moses  Webster  for  93  bushels  of  corn,  at 
$  1.25  per  bushel  ?  Ans.  2324-  bushels. 

19.  How  many  bushels  of  salt,  at  $  1.30  per  bushel,  must 
be  given  in  exchange  for  75  bushels  of  wheat,  at  $  1.25 
per  bushel  ?  Ans.  72/^  bushels. 

20.  If  a  sportsman  spend  ^  of  his  time  in  smoking,  £  in 
"  gunning,"  2  hours  per  day  in  loafing,  and  6  hours  in 
eating,    drinking,    and  sleeping,  how  much  remains  for 
useful  purposes  ?  Ans.  2  hours. 

21.  If  a  lady  spend  £  of  her  time  in  sleep,  -£  in  making 
calls,  ^  at  her  toilet,    y  in  reading  novels,  and  2  hours 
each  day  in  receiving  visits,  how  large  a  portion  of  her 
time  will  remain  for  improving  her  mind,  and  domestic 
employments  ?  Ans.  3f  £  hours  per  day. 

22.  What  will  a  piece  of  land  7§  rods  long,  and  5|  rods 
wide,  come  to  at  $  25.75  per  acre  ?      Ans.  $6.65|££. 

23.  If  5f  ells  English  cost  $  15.16,  what  will  71£  yards 
cost  ;  Ans.  $  155.39. 

24.  If  a  staff  4  feet  long  cast  a  shadow  5f  feet,  what  is 
the  height  of  that  steeple  whose  shadow  is  150  feet  ? 

Ans.  107|  feet. 


S£CT.51.]        MISCELLANEOUS    QUESTIONS.  167 

25.  Borrowed  of  James  Day  $  150  for  six  months  ;  after 
wards  I  lent  him  $  100  ;   how  long  shall  he  keep  it  to 
indemnify  him  for  the  sum  he  lent  me  ?     Ans.  9  months. 

26.  A  certain  town  is  taxed  $  6045.50  ;  the  valuation  of 
the  town  is  $293275.00  ;  there  are  150  polls  in  the  town, 
which  are  taxed  $  1.20  each.     What  is  the  tax  on  a  dol 
lar,  and  what  does  A.  pay,  who  has  4  polls,  and  whose 
property  is  valued  at  $  3675  ? 

Ans.  $  0.02.     A.'s  tax  $  78.30. 

27.  What  is  the  value  of  97  pigs  of  lead,  each  weighing 
2cwt.  3qr.  lllb.,  at  £3.  17s.  9d.  per  cwt.  ? 

AI.S.  £1074.  Os.  6£fcd. 

28.  What  is  the  interest  of  $  17.86,  from  Feb.  9,  1840, 
to  Oct.  29,  1842,  at  7£  per  cent.  ?         Ans.  $  35.24-}-. 

29.  What  is  the  interest  of  $97.87,  from  Jan.  7,  1840, 
to  Sept.  25,  1842,  at  9  per  cent.  ?         Ans.  $  23.92+. 

30.  T.  Jones'  note  for  $  1728  is  dated  March  1,  1836  ; 
Sept.  25,  1836,  was  received  $  50.00, 
Jan.  1,  1837,                do.  $60.00, 
June  7,  1837,               do.  $  8.00, 
Dec.  25,  1837,             do.                                        $  10.00, 
March  6,  1838,            do.  $5.00, 
Sept.  1,  1838,              do.                                          $9.00, 
Jan.  1,  1839,                do.                                        $300.00, 
July  4,  1839,                do.                                        $  100.00, 
Sept.  6,  1840,               do.                                          $  14.00, 
Jan.  25,  1841,               do.                                        $500.00, 
Dec.  11,  1841,             do.                                          $  15.00, 
March  9,  1842,            do.                                        $  200.00, 
What  is  due  Nov.  29,  1842  ?                     Ans.  $  1060.29. 

31.  $  1000.  Salem,  N.  H.,  Oct.  29,  1836. 
For  value  received,  I  promise  to  pay  Luther  Emer 
son,  Jr.,  or  order,  on  demand,  one  thousand  dollars  with 
interest.  Emerson  Luther. 

Attest,  Adams  Ayer. 

On  this  note  are  the  following  indorsements. 
Jan.  1,  1837,  was  received  $  125.00, 

June  5,  1837,  do.  $31600, 

Sept.  25,  1837,  do.  $  417.00, 

April  1,  1838,  do.  $  100.00, 

July  7,  1838,  do.  $  50.00 ; 

What  is  due,  at  compound  interest,  Oct.  29,  1842 

Ans.  $53.79. 


168  MISCELLANEOUS    QUESTIONS.         [SECT.  61. 

32.  J.  Ladd's  garden  is  100  feet  long  and  80  feet  wide; 
he  wishes  to  enclose  it  with   a  ditch  4  feet  wide;  how 
deep   must  it  be  dug,   that  the  soil  taken  from  it  may 
raise  the  surface  one  foot.  Ans.  5^£  feet. 

33.  How  many  yards  of  paper,  that  is  30  inches  wide, 
will  it  require  to  cover  the  walls  of  a  room,  that  is  15£ 
feet  long,  ll£  feet  wide,  and  7f-  feet  high  ? 

Ans.  55££  yards. 

34.  Charles  Carleton  has  agreed  to  plaster  the  above 
room  at    10  cents  per  square  yard  ;    what  will  be  his 
bill  ?  Ans.  $6.54f. 

35.  How  many  cubic   inches  are  contained  in  a  cube, 
that  may  be  inscribed  in  a  sphere  40  inches  in  diame 
ter  ?  Ans.  12316.8-)-  inches. 

36.  The  dimensions  of  a  bushel  measure  are  1H£  inches 
wide,   and  8  inches  deep  ;    what  should  be  the  dimen 
sions  of  a  similar  measure,  that  would  contain  4  quarts  ? 

Ans.  9£  inches  wide,  4  inches  deep. 

37.  A  gentleman  willed  ^  of  his  estate  to  his  wife,  and  £ 
of  the  remainder  to  his  oldest  son,  and  £  of  the  residue, 
which  was  $-151.33},  to  his  oldest  daughter  ;   how  much 
of  his  estate  is  left  to  be  divided  among  his  other  heirs  ? 

Ans.  $  756.66f. 

38.  A  man  bequeathed  £  of  his  estate  to  his  son,  and  £ 
of  the  remainder  to  his  daughter,  and  the  residue  to  his 
wife  ;    the    difference    between    his  son  and  daughter's 
portion  was  $  100  ;  what  did  he  give  his  wife  ? 

Ans.  $  600.00. 

39.  A  young  man  lost  J  of  his  capital  in  speculation  ; 
he  afterwards  gained  $500  ;  his  capital  then  was  $  1250; 
what  was  the  sum  lost  ?  Ans.  $250.00. 

40.  From  ^  of  a  yard,  there  was  sold  £  of  it  ;  how  much 
remained  ?  Ans.  •£$  yard. 

41.  Sold  a  lot  of  shingles  for  $  50,  and   by  so  doing  I 
gained  12£  per  cent.  ?  what  was  their  value  ? 

Ans.  $  44.44£. 

42.  If  tallow  be  sold  at  7£d.  per  lb.,  what  is  the  value 
of  17cwt.  3qr.  181bs.  ?  Ans.  $208.95§. 

43.  If  T3T  of  a  yard  cost  $5.00,  what  quantity  will  $  17.50 
purchase  ?  Ans.  f £  yard. 

44.  If  a  man  travel  17rd.   10ft.  in  T7T  of  an  hour,  how 
far  will  he  travel  in  8£  hours  ? 

Ans.  1  mile,  928f  feet. 


SECT.  51.]         MISCELLANEOUS    QUESTIONS.  1(39 

45.  When  $  11.75  are  paid  for  2f-  acres,  what  quantity 
will   8  100.00  purchase  ?  Ans.  19A.  1R.  3vMf>fp. 

46.  John  Savory  and  Thomas  Hardy  traded  in  company; 
Savory  put  in  for  capital  $  1000  ;  they  gained  $  123.00  ; 
Hardy  received  for  his  share  of  the  gains  $  70  ;   what 
was  his  capital  ?  Ans.  $  1206.89£f . 

47.  E.  Fuller  lent  a  certain  sum  of  money  to  C.  Lam- 
son,  and,  at  the  end  of  3  years,  7  months,  and  20  days, 
he  received  interest  and  principal  $  1000  ;   what  was  the 
sum  lent  ?  Ans.  $  820.79f  jj f 

48.  Lent  $  88  for  18  months,  and  received  for  interest 
and  principal  $97.57  ;  what  was  the  per  cent.  ? 

Ans.  7|-  per  cent. 

49.  When  £  of  a  gallon  cost  $  87,  what  cost  7£  gallons  ? 

Ans.  $  1051.25. 

50.  When   871  are  paid  for  18^  yards  of   broadcloth, 
what  cost  5  yards  ?  Ans.  $  19.2(3^. 

51.  How  many  yards  of  cloth,  at  $  4.00  per  yard,  must 
be  given  for  IStons.  17cwt.  3qr.  of  sugar,  at  $9.50  per 
cwt.  ?  Ans.  897-^2-  yards. 

52.  How   much  grain,  at    $  1.25    per   bushel,  must  be 
given  for  98  bushels  of  salt,  at  §  0.45  per  bushel  ? 

Ans.  35^  bushels. 

53.  How  many  acres  of  land,  at  $  37.50  per  acre  must 
be  given  for  Stitons.  IScwt.  3qr.  201bs.  of  coal,  at   §8.50 
per  ton  ?  Ans.  19A.  2R.  33^p. 

54.  A  person,  being  asked  the  time  of  day,  replied,  that 
Y  of  the  time  passed  from  noon  was  equal  to  -^  of  the 
time  to  midnight.     Required  the  time. 

Ans.  40  minutes  past  4. 

55.  How  many  cubic  feet  of  water  in  a  pond,  that  con 
tains  200  acres,  and  is  20  feet  deep  ? 

Ans.  174,240,000  feet. 

56.  On  a  certain  night,  in  the  year  1842,  rain  fell  to  the 
depth   of  3  inches  in  the  town  of  Haverhill  ;  the  town 
contains  about  20,000  square  acres.     Required  the  num 
ber  of  hogsheads  of  water  fallen,   supposing  each  hogs 
head  to  contain  100  gallons,  and  each  gallon  282  cubic 
inches.  Ans.  1334G!)42hhd.  55gal.  Iqt.  Opt.  2£Sgi. 

57.  If  the  sun  pass  over  one  degree  in  4  minutes,  and 
the  longitude  of  Boston  is  71°  4'  west,  what  will  be  the 

o 


170  MISCELLANEOUS    QUESTIONS.         [SECT.  51. 

time  at  Boston,  when  it  is  llh.  10m.  A.  M.  at  London  ? 
Ans.  6h.  31m.  44sec.  A.  M. 

58.  When  it  is  2h.  36m.  A.  M.   at  the  Cape  of  Good 
Hope,  in  longitude  18°  24'  east,   what    is   the    time  at 
Cape  Horn,  in  longitude  67°  21'  west  ? 

Ans.  Sh.  53m.  P.  M. 

59.  Yesterday  my  longitude,   at  noon,  was  16°  IS'  west ; 
to-day  I  perceive  by  my  watch,  which  has  kept  correct 
time,    that   the  sun  is  on  the  meridian   at    llh.    30m.  ; 
what  is  my  longitude  ?  Ans.  22°  18'  west. 

60.  Sound,  uninterrupted,  will  pass  1142  feet  in  one  sec 
ond,  how  long  will  it  be  in  passing  from  Boston  to  Lon 
don,  the  distance  being  about  3000  miles  ? 

Ans.  3h.  51m.  10-J-sec. 

61.  The  time  which  elapsed  between  seeing  the  flash  of 
a  gun,  and  hearing  its  report,  was  10  seconds  ;    what 
was  the  distance  ?  Ans.  2  miles.  800  feet. 

62.  If  a  globe  of  silver,  2  inches  in  diameter,  be  worth 
$  125,  what  would  be  the  value  of  a  globe  3  inches  in 
diameter?  Ans.  $  42l.87£. 

63.  J.  Pearson  has  tea,  which  he  barters  with  M.  Swift, 
at  10  cents  per  Ib.  more  than  it  costs  him,  against  sugar, 
which  costs  Swift  15  cents  per  Ib.,  but  which  he  puts  at 
20  cents  per  Ib.,  what  was  the  first  cost  of  the  tea  ? 

Ans.  $  0.30. 

64.  Q,.  and  Y.  barter  ;  Q,.  makes  of  10  cents  12£  cents  ; 
Y.  makes  of  15  cents  19  cents  ;  who  makes  the  most 
per  cent.,  and  by  how  much  ? 

Ans.  Y.  makes  If  per  cent,  more  than  Q,. 

65.  A  certain  individual  was  born  in  1786,  September  25, 
at  27  minutes  past  3  o'clock,  A.  M.,  how  many  minutes 
old  will  he  be  July  4,  1844,  at  30  minutes  past  5  o'clock, 
P.  M.  ?  Ans.  30,386,283  minutes. 

66.  The  longitude  of  a  certain  star  is  3s.  14°.  26'.   14"., 
and  the  longitude  of  the  moon   at  the  same  time  is  8s. 
19°.  43'  28".,  how  far  will  the  moon  have  to  move  in  her 
orbit  to  be  in  conjunction  with  the  star  ? 

Ans.  6s.  24°.  42'.  46". 

67.  From  a  small   field  containing  3A.  1R.  23p.  200ft., 
there  were  sold  1A.  2R.  37p.  30yd.  8ft.  ;   what   quantity 
remained  ?  Ans.  1A.  2R.  25p.  21yd.  5ft.  36in. 

68.  What  part  of  f  of  an  acre  is  f  of  an  acre? 

Ans. 


SECT. 51.]        MISCELLANEOUS    QUESTIONS.  J71 

69.  My  chaise  having  been  injured  by  a  very  bad  boy,  I 
am  obliged  to  sell  it  for  $  68.75,  which  is  40  per  cent, 
less  than  its  original  value,  what  was  the  cost  ? 

Ans.  $  114.58^. 

70.  Charles  Webster's  horse  is  valued  at   $  120,  but  he 
will  not  sell  him  for  less  than  $  134.40  ;  what  per  cent, 
does  he  intend  to  make  ?  Ans.  12  per  cent. 

71.  Three  merchants,   L.  Emerson,  E.  Bailey,  and   S. 
Curtiss  engaged  in  a  cotton  speculation.     Emerson  ad 
vanced    $  3600,    Bailey    8  4200    and    Curtiss    $  2200. 
They  invested   their  whole  capital  in   cotton,  for  which 
they  received  $  15000  in  bills  on  a  bank  in  New  Orleans. 
These  bills  were  sold  to  a  Boston  broker  at  15  per  cent, 
below  par,  what  is  each  man's  net  gain  ? 

Ans.  Emerson  $990.00.     Bailey  $1155.00.     Curtiss 
8605.00. 

72.  Bought  a  box  made  of  a  plank  3£  inches  thick.     Its 
length  is  4ft.  9in.,  its  breadth  3ft.  7in.,  and  its  height  2ft. 
llin.     How  many  square  feet  did  it  require  to  make  the 
box,  and  how  many  cubic  feet  does  it  contain  ? 

Ans.  70/5-  square  feet,  29£  cubic  feet, 

73.  How  many   bricks  will  it  require  to  construct   the 
walls  of  a  house,  64  feet  long  and  32  feet  wide,  and  28 
feet  high  ;  the  walls  are  to  be  1ft.  4in.  thick,   and  there 
are   also  three  doors  7ft.  4in.  high,  and  3ft.  Sin.   wide  ; 
also  14  windows  3  feet  wide  and  6  feet  high,  and  16  win 
dows  2ft.  Sin.  wide  and  5ft.  Sin.  high.     Each  brick  is  to 
be  8  inches  long,  4  inches  wide,  and  2  inches  thick. 

Ans.  167,480  bricks. 

74.  John  Brown  gave  to  his  three  sons,  Benjamin,  Samuel, 
and  William,   $  1000  to  be  divided   in  the  proportion  of 
•£,  £,  and  ±  respectively  ;   but  William,  having  received 
a  fortune  by  his  wife,  resigns  his  share  to  his   brothers. 
It  is  required  to  divide  the  whole  sum  between  Benja 
min  and  Samuel. 

Ans.  Benjamin  $  571. 42f     Samuel  8  428.57f 

75.  Peter  Webster  rented  a  house  for  one  year  to  Thomas 
Bailey  for  $  100  ;   at  the  end  of  four  months,    Bailey 
rented  one  half  of  the  house  to  John  Bricket,  and  at  the 
end  of  eight  months,  it  was   agreed   by  Webster   and 
Bailey  to   rent  one   third  of  the   house  to  John  Dana 
What  share  of  the  rent  must  each  pay  ? 

Ans.  Webster  $61£,  Bailey  $27£,  and  Dana  $  ll£. 


172  MISCELLANEOUS    QUESTIONS.        [SECT.51. 

76.  Bought  365  yards  of  broadcloth,   for  which  I  paid 
,£576.  17s.  9d.  ;   for  how  much  must  the   cloth   be  sold 
per  yard  to  gain  25  per  cent.        Ans.  £  1.  19s.  62495^d. 

77.  John   Brown's  house   is   40  feet  square  ;    the   roof 
comes  to  a  point  over  the  centre  of  the  house,  and  this 
point  is  12  feet  above  the   garret  floor.     Required  the 
length  of  a  rafter,  which  extends  from  one  of  the  cor 
ners  of  the  house  to  the  highest  part  of  the  roof. 

Ans.  30.72-f-  feet. 

78.  Minot  Thayer  sold  broadcloth  at  84.40  per  yard,  and 
by  so  doing  he  lost  12  per  cent.  ;   whereas  he  ought  to 
have   gained   10  per  cent.     For  what   should  the   cloth 
have  been  sold  per  yard  ?  Ans.  8  5.50. 

79.  John   Crowell   sold   cloth  at    85.50   per  yard,    and 
gained   10  per  cent.  ;    whereas,   the  cloth  having  been 
damaged,  he  should  have  sold  it  12  per  cent,  less  than 
the  cost.     What  in  justice  should  he  have  charged  per 
yard  ?  Ans.  $  4.40.  . 

80.  Jacob  How  has  cloth,  which  he  purchased  for  12  per 
cent,  less  than  its  value  ;   but  he   sells  it  at  10  per  cent, 
more  than  it  is  worth,  and  by  so  doing  he  gains  $  1.10  on 
each  yard.     What  per  cent,  did  he  make  on  his  pur 
chase  ?  Ans.  25  per  cent. 

81.  A  gentleman  has  five  daughters,  Emily,  Jane,  Betsey, 
Abigail,    and    Nancy,    whose   fortunes    are    as    follows. 
The  first  two   and  the  last  two  have  $  19,000  ;  the  first 
four   $  19,200  ;  the  last  four  8  20,000  ;  the  first  and  the 
last  three  820,500  ;   the  first  three  and  the  last  $21,300. 
What  was  the  fortune  of  each  ? 

Ans.    Emily   has   8  5,000  ;    Jane    8  4,500  ;    Betsey 
$6,000  ;  Abigail  $  3,700  ;  and  Nancy  85,800. 


APPENDIX 


CANCELLING   METHOD. 

BY  the  Cancelling  Method  the  scholar  is  enabled  to  solve 
many  questions  with  less  than  half  the  labor,  that  would  be 
required  by  the  usual  process.  It  cannot,  however,  be  ap 
plied  to  all  the  rules  of  arithmetic,  nor  to  all  the  questions  under 
any  one  rule  ;  but  it  is  generally  used  in  the  operations  of 
those  questions  which  require  Multiplication  and  Division. 
The  system  is  not  new.  It  has  been  before  the  public  in 
some  form  or  other  for  centuries.  John  Birks,  who  published 
the  second  edition  of  his  most  excellent  system  of  "  Arithmet 
ical  Collections  "  in  London,  1764,  has  made  many  improve 
ments  in  the  system.  Since  that  period,  but  little  advance  has 
been  made  in  it.  Whether  the  author  has  made  his  system 
more  plain  and  intelligible  than  has  been  done  by  others,  the 
candid  public  must  judge.  He  has  spared  no  pains  to  exhibit 
its  applicability  and  utility  to  those  departments  of  arithmetical 
science  where  it  can  be  advantageously  employed.  He  be 
lieves  the  system  can  be  of  but  little  use  to  the  pupil,  until  he 
can  perform  the  questions  by  the  common  method.  Hence 
the  propriety  of  deferring  attention  to  this  method,  until  the 
common  rules  of  arithmetic  are  thoroughly  understood. 

GENERAL  RULE. 

1.  Equal  divisors  and  dividends  cancel  each  other. 

2.  When  the  product  of  two  divisors  is  equal  to  the  product 
of  two  dividends,  they  cancel  each  other. 


174  CANCELLING  METHOD. 

I.    Cancelling  applied  to  Compound  Fractions. 

RULE  1.  —  If  there  be  numbers  in  the  numerators  and  de 
nominators,  that  be  alike,  an  equal  number  of  the  same  value 
may  be  cancelled. 

1.  Reduce  §  of  \  of  \  of  \  of  |  to  a  simple  fraction. 

STATEMENT.  CANCELLED. 

2x3x4x7x8        2X#X4X7X$        ^ 

~ 


3x4x5x8x9  ~~#xx5x$X9~  45 

In  this  question,  we  find  a  3,  4,  and  8  among  the  numerators, 
and  also  the  same  numbers  among  the  denominators.  These 
we  cancel  before  we  commence  the  operation. 

2.  What  is  the  value  of  \  of  &  of  \\  of  %  ? 

OPERATION.  We  find  in  this  question, 

7X  $  XXXxXtf  _  7  8,  11,  and  17  among  the  nu- 

&XXfxX%  X  19  —  TJ?  merators,  also  the  same  num 

bers  among  the  denominators. 

These  we  cancel. 

3.  What  is  the  value  of  1  of  &  of  g  of  &  of  {$  of  825. 


7X  g  XtfX  7  Xfl*X25  __  1225  ?? 

"   "   170  = 


4.  Reduce  T6T  of  £|  of  £f  of  J|  of  4|  to  a  simple  fraction. 

5 

=  -  =  H  Ans. 


4        4 

5.  Required  the  value  of  |  of  ^  of  {%  of  ||  of  40. 

7X0X^0X^X40  _  280  _ 
"0X70X13X2^"     =  24  ^ 

6.  Reduce  {I  of  f|  of  2f  to  its  equivalent  value. 


7.  A^rhat  is  the  value  of  ,\  of  1|  of  £  of  3|  of  $  18  ? 


XXX  t  X^0X  #  X  1 


CANCELLING  METHOD.  175 

8.  What  is  the  value  of  A  of  J}  of  §J  of  $  7|  ? 
#_7 

--- 


9.  What  is  f  of  &  of  f  J  of  3|  gallons  ? 
4X  0  XWXW       4 


RULE  2,  —  When  there  are  any  two  numbers,  one  in  the  nu 
merators,  and  the  other  in  the  denominators,  which  may  be  di 
vided  by  a  number  without  a  remainder,  the  quotients  arising 
from  such  division  may  be  used  in  the  operation  of  the  ques 
tion,  instead  of  the  original  numbers.  The  quotients  also  may 
be  cancelled,  as  other  numbers. 

1.  Reduce  f  of  ||  of  ||  of  T\  to  its  lowest  terms. 

OPERATION.  In  performing  this  question, 

271  we    find    that   14  among  the 

56  numerators,  and  7  among  the 

=  495  AnS'  denominators>  maybe  divided 
by  7,  and  that  their  quotients 
will  be  2  and  1.  We  write 

the  2  above  the  14,  and  1  below  the  7.  We  also  find  a  21 
among  the  numerators,  and  a  27  among  the  denominators,  which 
may  be  divided  by  3,  and  that  their  quotients  will  be  7  and  9. 
We  write  the  7  above  the  21,  and  9  below  the  27.  We  again 
find  a  5  among  the  numerators,  and  a  25  among  the  denomi 
nators,  which  may  be  divided  by  5,  and  that  their  quotients 
will  be  1  and  5.  We  write  the  1  over  the  5,  and  the  5  below 
the  25.  We  then  multiply  the  4,  2,  7,  and  1  together  for  a 
numerator  =  56,  and  the  1,  9,  5,  and  11  for  a  denominator  = 
495.  The  answer  will  therefore  be  ^. 

2.  Reduce  Jf  of  ||  of  ^  of  £  to  a  simple  fraction. 

2621 

_    ^4 

"275     nS* 


176  CANCELLIJSG  METHOD. 

3.  What  is  the  value  of  f  of  |5  of  ft  of  if  of  $  34  ? 
1  3       $      & 

_W  . 

="    =  =  $6'75  Ans* 


1441 

NOTE.    The  above  rule  will  apply,  when  the  product  of  several  numbers  is 
to  be  divided  by  the  product  of  other  numbers. 

4.  What  is  the  continued  product  of  8,  4,  9,  2,  12,  16,  and  5 
divided  by  the  continued  product  of  40,  6,  6,  3,  8,  4,  and  20? 
1 


The  product  of  4  and  9  in  the  upper  line  is  equal  to  the 
product  of  6  and  6  in  the  lower,  therefore  they  are  cancelled  ; 
and  the  product  of  2  and  12  in  the  upper  line  is  equal  to  the 
product  of  3  and  8  in  the  lower  line  ;  also  the  product  of  16 
and  5  in  the  upper  line  is  equal  to  the  product  of  4  and  20  in 
the  lower  line  ;  these  are  all  cancelled.  We  also  find,  that  the 
8  in  the  upper  line  and  the  40  in  the  lower  line  may  be  divid 
ed  by  8,  and  their  quotients  will  be  1  and  5.  We  write  the  1 
above  the  8  and  the  5  below  the  40.  By  the  usual  process,  we 
now  find  our  answer  is  f. 

5.  What  is  the  continued  product  of  12,  13,  14,  15,  16,  18, 
20,  21,  and  24,  divided  by  the  continued  product  of  2,  3,  4,  5, 
6,  7,8,  9,  10,  and  11? 

3  2322272 

26208 


11111111      1 

IT.  In  finding  the  common  multiple  of  two  or  more  num 
bers,  any  one  number  that  will  measure  another  may  be 
cancelled. 

1.  What  is  the  least  common  multiple  of  4,  6,  8,  12,  16, 
10,  and  20  ? 

4)  4  0  $   12   16  1'0  20 

-o—  T-     -«-     4X3X4X5  =  240  Ans. 
34  5 


CANCELLING  METHOD.  177 

By  examining  this  question,  we  find  that  8  may  be  divided 
by  4,  12  by  6,  16  by  8,  and  20  by  10  ;  therefore  we  cancel  4, 
6,  8,  and  10. 

2.  What  is  the  least  common  multiple  of  5,  15,  30,  7,  14, 

and  28  ? 


In  this  question,  we  find  that  15  may  be  measured  by  5,  30 
by  15,  14  by  7,  and  28  by  14  ;  we  therefore  cancel  5,  15,  7, 
and  14. 

3.  What  is  the  least  common  multiple  of  1,  2,  3,  4,  5,  6,  7, 
8,  and  9  ? 

2)  ^##^56789     2X3X5X7X4X3  =  2520 
3)  5  3  7  4  9  [Ans. 

5  1  7T~3 

4.  What  is  the  least  common  multiple  of  9,  8,  12,  18,  24, 
36,  and  72  ? 

0  0  it  *$  M  n  72         72  Ans. 

5.  What  is  the  least  number  that  18,  24,  36,  12,  6,  20,  and 
48  will  measure  ? 

4)  it  U  36  a#  0  20  48     4X3X3X5X4  =  720  Ans. 
3)9  _  5    12 
3  5~T 


III.    SINGLE  PROPORTION, 

PERFORMED   BY   CANCELLING. 

RULE.  —  When  the  first  and  second  terms,  or  the  first  and 
third  terms,  can  be  divided  by  any  number  without  a  remain 
der,  their  quotients  may  be  used  in  the  operation  of  the  ques 
tions  instead  of  the  terms  themselves. 


178  CANCELLING    METHOD. 

1.  If  14cwt.  of  logwood  cost  $  56,  what  cost  95cwt.  ? 

OPERATION    BY   PROPORTION.  CANCELLING. 

CWt.     CWt.  8  A 

14  :  95  :  :  56 


__56  —£^  =  $  380  Ans. 

570  z* 

475  1 

14)5320($380  Ans. 
42 

112 
112 

0 

2.  If  23  men,  in  one  month,  can  dig  a  ditch  19  rods  long, 
8  feet  wide,  and  3  feet  deep,  how  many  men  would  it  require 
to  dig  a  ditch  57  rods  long,  4  feet  wide,  and  6  feet  deep,  in  the 
same  time  ? 

BY  PROPORTION.  CANCELLING. 

19x8x3  :  57X4X6  ::  23        3      1    $ 
J?  _4  #7X4X0X23 

152  228  *0xgxa      =  69  rmen> 

_3  _6  1*1  tAnS* 

456  1368 

23 
4104 
2736_ 

456)314~64(69  men,  Ans. 
2736_ 

4104 
4104 

3.  If  7  pairs  of  shoes  will  pur-  7 
chase  2  pairs  of  boots,  how  many          40X2 

pairs  of  boots   may  be   purchased          — = —  =  14  pairs, 
with  49  pairs  of  shoes  ?  [Ans. 

4.  If  a  staff  4  feet  in  length  cast  24 
a  shadow  6  feet  long,  how  high  is       4X^44 

that  steeple  whose  shadow  is  144      — ^ —  =  96  feet, 
feet?  [Ans. 


CANCELLING  METHOD. 


179 


5.  If  4  gallons   of  vinegar   be 
worth  9  gallons  of  cider,  how  many 
gallons  of  cider  will  it  require  to 
purchase  36  gallons  of  vinegar  ? 

6.  If  a  man  travel  765  miles  in 
75  days,  how  far  would  he  travel 
in  15  days  ? 


7.  If  15  yards  of  cloth,  that  is  3 
quarters  of  a  yard  wide,  are  suffi 
cient  to  make  a  garment,  how  many 
yards   will  it  require   to   line   the 
same  that  is  5  quarters  of  a  yard 
wide  ? 

8.  When  8200.85  are  paid  for 
39  barrels  of  flour,  what  must  be 
paid  for  13  barrels  ? 


9 
30X9 


l 


81  gallons, 
[Ans. 


153 


=  153  miles, 
[Ans. 


=  9  vards, 
'[Ans. 


66.95      1 


30 
3 


=  $66.95 
[Ans. 


IV.    COMPOUND  PROPORTION. 


PERFORMED   BY   CANCELLING. 


1.  If  a  man  travel  117  miles  in  30 
days,  employing  only  9  hours  a  day, 
how  far  would  he  go  in  20  days,  trav 
elling  12  hours  a  day  ? 


OPERATION. 


30 


20 


4 
13 


104  miles,  Ans. 


In  performing  this  question,  we  arrange  the  numbers,  that 
would  be  the  second  and  third  terms  in  the  regular  statement 
of  the  question  on  the  right  hand  of  a  perpendicular  line,  and 
the  numbers,  that  would  be  the  first  term,  on  the  left.  We  then 
divide  the  product  of  the  uncancelled  numbers  on  the  right  by 
the  product  of  the  uncancelled  numbers  on  the  left. 


180 


CANCELLING   METHOD. 


2.  If  6  men  in  16  days  of  9  hours 
each  build  a  wall  20  feet  long,  6  feet 
high,  and  4  feet  thick,  in  how  many 
days  of  8  hours  each  will  24  men 
build  a  wall  200  feet  long,  8  feet 
high,  and  6  feet  thick  ? 


3.  If  $  100  gain  8  6  in  12  months, 
how  much  would  $  800  gain  in  8 
months  ? 


4.  If  $  100  gain  $  6  in  12  months, 
what  must  be  the  sum  to  gain  $  16  in 
8  months  ? 


$400  Ans. 


5.  How  long  will   it  take  $  600     000 1  £00 
to  gain   $  12,  if  $  100  gain  $  6  in 
12  months  ? 


4  months,  Ans 


6  If  $  600  gain  $  18  in  6  months,     000 
what  is  the  rate  per  cent.  ?  0 


100 


6  per  cent.  Ans. 


7.  If  12  men  in  15  days  can  build     I 
a  wall  30  feet  long,  6  feet  high,  and 

3  feet  thick,  when  the  days  are  12 
hours  long,  in  what  time  will  60  men 
build  a  wall  300  feet  long,  8  feet 
high,  and  6  feet  thick,  when  they 
work  only  8  hours  a  day  ? 

8.  If  8  men  spend  $  32  in  13  weeks, 
what  will  24  men  spend  in  52  weeks  ? 


00 1    t£ 
#0  #00 


12 


10 


120  days,  Ans. 

3 

£g  4 
32 


$  384  Ans. 


CANCELLING   METHOD. 


181 


9.  If  16  horses  consume  84  bush 
els  of  grain  in  24  days,  how  many 
bushels  will  suffice  32  horses  48 
days? 


84 


10.  If  the  carriage  of 
5cwt.  3qr.,  150  miles  cost 
$  24.58,  what  must  be 
paid  for  the  carriage  of 
7cwt.  2qr.  251bs.,  64  miles 
at  the  same  rate  ? 


161 
30 


11.  If  7oz.  5dwt.  of 
bread  be  bought  at  4|d., 
when  corn  is  4s.  2d. 
per  bushel,  what  weight 
of  it  may  be  bought  for 
Is.  2d.,when  the  price 
per  bushel  is  5s.  6d.  ? 


33 
.19 


336  bushels,  Ans. 

173 
16 


24.58 


68037.44 
4830 


$  14.08+ 
[Ans. 


00  p0 


145 
2030 


=  lib.  4oz.  Sgdwt. 
[Ans. 


V.    CANCELLING  APPLIED  TO  THE  CHAIN  RULE. 

The  Chain  Rule  consists  in  joining  many  proportions  to 
gether  ;  and  by  the  relations  which  the  several  antecedents 
have  to  their  consequents,  the  proportion  between  the  first 
antecedent  and  the  last  consequent  is  discovered. 

This  rule  may  often  be  abridged  by  cancelling  equal  quanti 
ties  on  both  sides,  and  abbreviating  commensurables. 

NOTE.  The  first  numbers  in  each  part  of  the  question  are  called  ante 
cedents,  and  the  following,  consequents. 

1.  If  20  Ibs.  at  Boston  make  23  Ibs.  at  Antwerp,  and  150  Ibs. 
at  Antwerp  make  180  Ibs.  at  Leghorn,  how  many  pounds  at 
Boston  are  equal  to  144  Ibs.  at  Leghorn  ? 

OPERATION  BY  THE  CHAIN  RULE. 

20  Ibs.  of  Boston  =:  23  Antwerp, 
150  Ibs.  of  Antwerp  =  180  Leghorn, 
144  Ibs.  of  Leghorn. 


182 


CANCELLING  METHOD. 


4140)446400(107|§  Ibs.  Ans. 
4140 


ISO) 


32400 

28980 

3420 
4140 


19 
23 


It  will  be  per 
ceived  in  this 
operation,  that 
the  continued 
product  of  the 
antecedents  is 
divided  by  the 
continued  pro 
duct  of  the  con 
sequents. 

Hence  the  fol 
lowing 


RULE. —  Write  the  numbers  alternately,  that  is,  the  antece 
dents  at  the  left  hand,  and  the  consequents  at  the  right  hand  ; 
and,  if  the  last  number  stands  at  the  left  hand,  multiply  the 
numbers  of  the  left  hand  column  continually  together  for  a 
dividend,  and  those  at  the  right  hand  for  a  divisor ;  but,  if 
the  last  number  stands  at  the  right  hand,  multiply  the  numbers 
at  the  right  hand  column  continually  together  for  a  dividend, 
and  those  at  the  left  for  a  divisor ;  and  the  quotient  will  be 
the  answer. 

OPERATION    BY    CANCELLING. 

23 
155 

16 


2480 


=  1074?  Ibs.  Ans. 


2.  If  12  Ibs.  at  Boston  make  10  Ibs. 
at  Amsterdam,  and  10  Ibs.  at  Amster 
dam  make  12  Ibs.  at  Paris,  how  many 
pounds  at  Boston  are  equal  to  80  Ibs. 
at  Paris  ? 

3.  If  25  Ibs.  at  Boston  are  equal  to 
22  Ibs.  at  Nuremburg,  and  88  Ibs.  at 
Nuremburg   are  equal   to  92  Ibs.    at 
Hamburg,  and  46  Ibs.  at  Hamburg  are 
equal  to  49  Ibs.  at  Lyons,  how  many 
pounds  are  equal  to  98  Ibs.  at  Lyons  ? 


80 

80"  Ibs.  Ans. 

25 

$$  4 
40 

n  $ 

100  Ibs.  Ans. 


CANCELLING    METHOD. 


183 


4.  If  24  shillings  in  Massachu 
setts  are  equal  to  32  shillings  in 
New  York  ;  and  if  48  shillings  in 
New  York  are  equal  to  45  shillings 
in  Pennsylvania ;  and  if  15  shil 
lings  in  Pennsylvania  are  equal  to 
10  shillings  in  Canada ;  how  many 
shillings  in  Canada  are  equal  to 
100  shillings  in  Massachusetts  ? 


10 
100 


1000 
12 


=  83is.  Ans. 


5.  If  17  men  can  do  as  much 
work  as  25  women,  and  5  women 
do  as  much  as  7  boys,  how  many 
men  would  it  take  to  do  the  work 
of  75  boys  ? 


17 


255 

— -= 36f  men,  Ans. 


6.  If  10  barrels  of  cider  will  pay 
for  5  cords  of  wood,  and  20  cords  of 
wood  for  4  tons  of  hay,  how  many 
barrels  of  cider  will  it  take  to  pur 
chase  50  tons  of  hay  ? 


Ans. 


7.  If  100  acres  in  Brad 
ford  be  worth  120  in  Ha- 
verhill,  and  50  in  Haver- 
hill  worth  65  in  Methuen, 
how  many  acres  in  Brad 
ford  are  equal  to  150  in 
Methuen  ? 


13 


o 
10 
25 


1250 


=  96/3  acres,  Ans. 


8.  If  10  Ibs.  of  cheese  are 
equal  in  value  to  7  Ibs.  of  but 
ter,  and  11  Ibs.  of  butter  to  2 
bushels  of  corn,  and  1 1  bushels 
of  corn  to  8  bushels  of  rye,  and 
4  bushels  of  rye  to  one  cord  of 
wood,  how  many  pounds  of 
cheese  are  equal  in  value  to  10 
cords  of  wood  ? 


11 
11 


3025 


=  432J  Ibs.  Ans. 


184  CANCELLING   METHOD. 

MISCELLANEOUS   QUESTIONS. 

1.  Required  the  number  of  cubic  feet  in  a  box,  2|  feet  wide, 
11  feet  high,  and  14^  feet  long  ? 

2i  =  l;  1I=V;  14*  =  W- 

0X^0X231        231 
____=     -T  =  5 

2.  What  cost  15$  yards  of  cloth,  2§  yards  wide,  at  $  3£  per 
square  yard  ? 

15|  =  T;  2§  =  i;  %  =  ?. 
41 

410 

-  T  -  *  136i  Ans* 

1 

3.  If  $  12£  will  purchase  a  piece  of  land  that  is  9|  rods 
long  and  6|  rods  wide,  how  long  a  piece  that  is  3|  rods  wido 
may  be  obtained  for  $  9|  ? 


4 

10    , 
=  12  rods' 


x 

4.  When  18f  square  rods  of  land  are  sold  for  $  3^,  what  is 
the  value  of  62|  square  rods  ? 


1  1 

JT  X  125X4*        125 


5.  How  many  boxes  that  are  1  foot  7  inches  high,  1  foot  5 
inches  wide,  and  5  feet  1  inch  long,  will  it  require  to  hold  the 
same  quantity  that  a  box  4  feet  9  inches  wide,  2  feet  10  inches 
high,  and  25  feet  5  inches  long,  would  contain  ? 

325 

30 
=  T  =  30  boxes'  Ans' 


END. 


CATALOGUE  OF 

APPROVED  SCHOOL  BOOKS, 

PUBLISHED  AND  SOLD  BY 

ROBERT    S.    DAVIS, 

NO.  77,  Washington  Street,  BOSTON. 


SOLD  ALSO  BY  THE  PRINCIPAL  BOOKSELLERS  IN  THE  UNITED  STATES. 


GREENLEAF'S   INTRODUCTION  TO  THE   NATIONAL  ARITHMETIC. 
GREENLEAF'S    NATIONAL  ARITHMETIC,   improved  stereotype  edit. 
GREENLEAF'S  COMPLETE  KEY  to  the  NATIONAL  ARITHMETIC. 
GREENLEAF'S  LESSONS  IN  PUNCTUATION,  5th  edition,  improved. 
SMITH'S    CLASS-BOOK   OF  ANATOMY,   7th  improved  stereotype  ed. 
CAESAR'S    COMMENTARIES,  with  English  Notes  by  F.  P.  Leverett. 
CICERO'S  ORATIONS,  with  English  Notes  by  Charles  Folsom,  stereo,  ed. 
FISK'S   GREEK  GRAMMAR,  Twenty-first  improved  stereotype  edition. 
FISK'S   GREEK   EXERCISES,  (adapted  to  the  Grammar,)  stereotype  ed. 
CLASSICAL   READER,  by  Greenwood  and  Emerson  ;    improved  stereo,  ed. 
BOSTON    SCHOOL  ATLAS,   14th  edition,  improved  and  stereotyped. 
ADAMS'  GEOGRAPHY   AND   ATLAS,  17th   ed.  revised  and   improved. 
WALKER'S  BOSTON  SCHOOL  DICTIONARY,  "Genuine  Boston  Ed." 
ALGER'S   MURRAY'S  GRAMMAR,  36th  improved  stereotype  edition. 
ALGER'S   MURRAY'S   EXERCISES,  18th  improved  stereotype  edition. 
ALGER'S  PRONOUNCING  INTRODUCTION  to  MURRAY'S  RKADER. 
ALGER'S   MURRAY'S  PRONOUNCING  ENGLISH  READER. 
PARKER'S  EXERCISES  IN  ENGLISH   COMPOSITION,  39th  edition. 
AIDS  TO   ENGLISH  COMPOSITION,  designed  as  a  SEQUEL  to  Par- 
ker's  Progressive  Exercises  in  English  Composition,  by  the  same  author. 
Ql3~  Also  constantly  on  hand,  (in  addition  to  his  own  publications,)  a  complete  as 
sortment  of  School  Books  and  Stationery,  which  are  offered  to  Booksellers,  School 
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BOOKS  FOR  SCHOOLS  AND  ACADEMIES, 

PUBLISHED  BY  ROBERT  S.  DAVIS,  BOSTON. 

GREENLEAF'S  INTRODUCTION  TO  THE  NATIONAL  ARITH 
METIC, 

On  the  Inductive  System,  combining  the  Analytic  and  Synthetic  Methods, 
with  the  Cancelling  System,  in  which  the  Principles  of  Arithmetic  are  ex 
plained  and  illustrated  in  a  familiar  manner  ;  designed  for  Common  Schools 
throughout  the  United  Stales  ;  by  BENJAMIN  GHEENLEAF,  A.  M,,  Prin 
cipal  of  Bradford  Teachers'  Seminary.  172  pages,  handsomely  printed  on 
fine  paper,  and  neatly  bound,  half  morocco  ;  Sixth  Edition,  revised,  im 
proved  and  stereotyped. 

GREENLEAF'S  NATIONAL  ARITHMETIC, 

Designed  for  the  more  advanced  scholars  in  Common  Schools  and  Acade 
mies,  forming  a  volume  of  upwards  of  300  pages,  handsomely  printed  on  fine 
paper,  and  strongly  bound  in  leather.  Sixteenth  Improved  Stereotype  Edi 
tion  ;  with  an  APPENDIX,  comprising  the"  Cancelling  Method." 

GREENLEAF'S  COMPLETE  KEY  (for  teachers  only.) 
B.  GREENLEAF,  Esq.  Dear  Sir  :  We  have  examined  your  Arithmetics, 
the  National  and  Introductory,  and  take  pleasure  in  expressing  to  you  our 
high  satisfaction  in  them,  as  superior  to  any  books  in  this  branch  of  educa 
tion  with  which  we  are  acquainted.  We  are  especially  pleased  with  the 
accuracy  and  precision  of  the  definitions,  and  with  the  clearness  and  fullness 
of  illustration  by  the  examples.  The  two  together  seem  to  be  just  what  are 
needed,  and  we  are  inclined  to  say  all  that  are  needed  on  this  subject  in  our 
Public  Schools.  In  accordance  with  this  view  of  your  books,  as  members  of 
the  General  School  Committee,  we  have  encouraged  their  use  in  the  schools 
in  this  town. 

(Signed,)  EDWAKD  A.  LAWKENCE,  )    Superintending 

A.  S.  TRAIN,  5  School  Committee. 

Haverfiill,  Mass.,  May  22,  1843. 

Phillips  Academy,  Andover,  Feb'y  10,  1844. 

We  have  adopted  the  National  Arithmetic  as  a  text-book  in  this  Institu 
tion.  Having  examined  most  of  our  popular  systems  of  Arithmetic,  I  can 
say  with  sincerity,  that  I  regard  your  book  as  better  adapted  to  meet  the 
wants  of  Academies,  and  the  higher  classes  in  Common  Schools,  than  any 
other  treatise  on  the  subject. 

(Signed,)  W.  H.  WELLS,  Inst.  English  Department. 

This  Arithmetic  is  also  the  regular  text-book  in  the  Normal  Schools  in 
Bridgewater  and  Lexington,  (Mass.,)  and  is  highly  recommended  by  the  dis 
tinguished  principals  of  those  Institutions,  viz.,  N.  Tillinghast,  Esq.,  and 
Rev.  Samuel  J.  May.  Greenleaf'p  Arithmetics,  (Introduction  and  National,) 
are  used  exclusively  in  most  of  the  Private  Schools,  and  Collegiate  and  Clas 
sical  Institutes  in  New  York  City,  and  have  been  extensively  adopted  in  all 
parts  of  the  United  States. 

GREENLEAF'S  LESSONS  IN  PUNCTUATION,  Fifth  edition. 

PARKER'S  PROGRESSIVE  EXERCISES  IN  ENGLISH 
COMPOSITION. 

$&•  The  reputation  of  this  invaluable  little  manual,  is  now  BO  well  estab 
lished,  that  it  is  deemed  unnecessary  to  present  any  of  the  flattering  testimo 
nials  in  its  favor,  from  the  many  distinguished  teachers,  in  all  parts  of  the 
United  States,  who  have  adopted  it  as  a  text-book. 

PARKER'S  AIDS  TO  ENGLISH  COMPOSITION, 

Designed  as  a  Sequel  to  Parker's  Progressive  Exercises  in  English  Com 
position,  prepared  for  Students  of  all  grades. 
ALGER'S  MURRAY'S  ENGLISH  GRAMMAR,  AND  EXERCISES. 


APPROVED  BOOKS  FOR  SCHOOLS  AND  ACADEMIES- 
SMITH'S  CLASS  BOOK  OF  ANATOMY, 

Explanatory  of  the  first  principles  of  Human  Organization,  as  the  basis 
of  Physical  Education  ;  with  numerous  Illustrations,  a  full  Glossary,  or  ex 
planation  of  technical  Terms,  and  practical  Questions  at  the  bottom  of  the 
page.  Designed  for  Schools  and  Families.  New  stereotype  edition. 

QC/-  This  work  has  received  the  highest  testimonials  of  approbation  from 
the  most  respectable  sources,  and  has  already  been  adopted  as  a  text-book, 
in  many  Schools  and  Colleges  in  the  United  States. 

FISK'S  GRAMMAR  OF  THE  GREEK  LANGUAGE,  new  edition. 

The  requisites  in  a  Manual  of  Grammar,  are  simplicity  and  lucidness  of 
arrangement,  condensation  of  thought,  and  accuracy  of  principle  and  ex 
pression.  These  requisites  Mr.  Fisk  appears  to  have  attained  in  a  consid 
erable  degree  in  his  Greek  Grammar,  of  which  we  have  expressed  approba 
tion  by  intoducing  it  into  our  Schools. 

FORREST   AND  WYCKOFF,  Principals  of  Collegiate   School,  New  York. 

OCr"  Fisk's  Greek  Grammar  is  used  in  Harvard  University,  and  in  many 
other  Collegiate  and  Academic  Institutions,  in  various  parts  of  the  United  States. 

FISK'S  GREEK  EXERCISES,  new  edition. 

Greek  Exercises;  containing  the  substance  of  the  Greek  Syntax,  illus 
trated  by  Passages  from  the  best  Greek  Authors,  to  be  written  out  from  the 
words  given  in  their  simplest  form  ;  by  BENJAMIN  FRANKLIN  FISK. 
"  Consuetudo  et  exercitatio  facilitatem  maxim  parit."  —  Quintil.  Adapted 
to  the  Author's  "  Greek  Grammar."  Sixteenth  stereotype  edition. 

Fisk's  Greek  Exercises  are  well  adapted  to  illustrate  the  rules  of  the 
Grammar,  and  constitute  a  very  useful  accompaniment  thereto. 

(Signed)         J.  B.  KlDDER,  Teacher  of  Select  School,  New  York. 

LEVERETT'S  CESAR'S  COMMENTARIES. 

Caii  Julii  Ccesaris  Commentarii  de  Bello  Gallico  ad  Codices  Parisinos 
recensiti,  a  N.  L.  Achaintre  et  N.  E.  Lemaire.  Accesserunt  Notula?  An- 
glicse,  atque  Index  Historicuset  Geographicus.  Curavit  F.  P.  LEVERETT. 

FOLSOM'S  CICERO'S  ORATIONS. 

M.  T.  Ciceronis  Orationes  Quaedam  Select*,  Notis  illustratae.  [By 
CHARLES  FOLSOM,  A.  M.]  In  Usum  Academiae  Exoniensis.  Editio 
stereotypa,  Tabulis  Analyticis  instructa. 

BOSTON  SCHOOL  ATLAS,  one  volume,  quarto. 
Embracing  a  Compendium  of  Geography.    Containing  eighteen  Maps  and 
Charts.    Embellished  with  instructive  Engravings.    Fifteenth  edition,  hand 
somely  printed  from  stereotype  plates,  and  the  Maps  are  beautifully  colored. 

ADAMS'S  SCHOOL  GEOGRAPHY,  AND  ATLAS. 

New  Edition,  improved  ;  being  a  Description  of  the  World,  in  three 
parts.  To  which  is  added  a  brief  Sketch  of  Ancient  Geography  ;  a  plain 
Method  of  Constructing  Maps  ;  and  an  Introduction  to  the  Use  of  the 
Globes.  Illustrated  by  numerous  Engravings.  Accompanied  by  an  7m- 
proved  Atlas.  Designed  for  Schools  and  Academies  in  the  United  States. 
By  DANIEL  ADAMS,  A.  M.,  Author  of  the  "  New  School  Arithmetic." 

THE  CLASSICAL  READER, 

A  Selection  of  Lessons  in  Prose  and  Verse,  from  the  most  esteemed  Eng 
lish  and  American  Writers.  Intended  for  the  use  of  the  higher  classes  in 
Public  and  Private  Seminaries.  By  Rev.  F.  W.  P.  GREENWOOD,  D.  D., 
and  G.  B.  EMERSON,  A.  M.,  of  Boston.  Tenth  stereotype  edition. 

$3"  R.  S.  D.  has  constantly  on  hand,  (in  addition  to  his  own  publications,) 
a  complete  assortment  of  School  Books  and  Stationery,  which  are  offered  to  Book 
sellers,  School  Committees,  and  Teachers,  wholesalt  and  retail,  on  liberal  terms. 


RECOMMENDATIONS    OF    GREENLEAF*S    ARITHMETIC. 

Benjamin  Greenleaf,  Esq.     Dear  Sir  :  I  regard  your  National  Arithmetic  as 
one  of  the  best  I  have  ever  seen.     Perhaps  the  best  proof  of  the  estimation  in 
which  I  hold  its  merits,  is  the  fact,  that  I  use  it  in  the  school  under  my  care. 
I  am,  Sir,  very  respectfully,  yours, 

ROGER  S.  HOWARD, 
Principal  of  the  Latin  High  School. 
Newburyport,  May  5,  1843. 

I  have  used  Mr.  Greenleaf 's  National  Arithmetic  in  my  School  for  nearly 
two  years  ;  and,  having  thus  tested  its  good  qualities,  I  can  cheerfully  recom 
mend  it,  as  a  system  of  arithmetic  well  adapted  for  giving  an  individual  a 
thorough  knowledge  of  the  science.  A.  H.  MERRIAM, 

Preceptor  of  Westminster  Academy. 

Westminster,  (Mass.)  June  6,  1843. 

I  have  made  use  of  Mr.  Greenleaf 's  National  Arithmetic  in  my  school,  and 
am  of  the  opinion,  that  it  possesses  superior  excellences  as  an  Arithmetic,  and 
well  adapted  to  our  common  and  higher  Schools. 

F.  G,  PRATT, 

Bridgewater,  (Mass.)  June  14,  1843.         Preceptor  of  Sridgewater  Academy. 

The  undersigned,  having  examined  the  National  Arithmetic  on  the  Inductive 
System,  by  Benjamin  Greenleaf,  Esq.,  do  not  hesitate  to  pronounce  it  a  work 
of  high  merit.  The  various  subjects  treated  of  in  it  are  arranged  in  a  manner 
at  once  philosophical  and  practical  ;  and,  in  the  opinion  of  the  undersigned,  it 
contains  a  greater  amount  of  useful  and  valuable  matter,  some  of  which  must 
otherwise  be  sought  for  in  rare  books,  than  any  other  similar  work  with  which 
they  are  acquainted.  And  they  cheerfully  recommend  it  to  teachers  and  learn 
ers,  as  a  work  of  high  and  undoubted  worth. 

THOMAS  C.  BAKER,     1 
JOHN  P.  PENDLETON,  1     Superintending 
JOHN  P.  ADAM,  f  School  Committee. 

A.  T.  C.  DODGE, 
Prospect,  (Me.)  March  1,  1843. 

Extract  from  a  Letter  from  Hiram  Orcutt,  Esq. ,  Teacher. 

Hebron,  N.  H.,  Feb.  27,  1843. 

"  Your  Arithmetic  I  have  had  opportunity  thoroughly  to  examine,  having 
introduced  it  into  my  School,  and  conducted  two  large  classes  of  teachers  entirely 
through  it.  And  I  can  freely  say,  Sir,  Miat  in  my  opinion,  no  book  of  the  kind 
now  extant,  is  so  well  calculated  to  lead  the  student  to  a  thorough  practical 
knowledge  of  figures  as  this." 

New  Bedford,  Mass.,  Dec.  26,  1842. 

Benjamin  Greenleaf,  Esq.  Dear  Sir  :  We  have  examined  your  Introductory 
Arithmetic,  and  are  much  pleased  with  the  plan  and  execution.  The  examples 
are  practical  ;  the  rules  clear  and  concise  ;  the  principles  of  the  science  are  un 
folded,  and  its  practical  uses  explained  with  great  perspicuity  and  simplicity. 
We  deem  it  eminently  calculated  to  answer  the  object  for  which  it  is  de 
signed. 

BENJAMIN  EVANS,  Principal  of  the  Charles-St.  School. 

EBENEZER  HERVEY,    do.  Sixth-St.        do. 

A.  L.  GLEASON,  do.  Bush-St.        do. 

WILLIAM  F.  Dow,         do.  William-St.    do. 

ALBERT  CONANT,  do.  Maxfield-St.  do. 

FRED.  F.  DEWEY,         do.  Hill  do. 

ADAM  MACKIE,  do.  Grove  do. 


RECOMMENDATIONS    OF    GREENLEAF's    ARITHMETIC. 

From  D.  P.  Page,  Esq.,  Principal  of  the  English   High  School,  Newburyport. 

Benjamin  Greenleaf,  Esq.  Dear  Sir  :  I  have  with  much  care  examined  the 
National  Arithmetic,  of  which  you  are  the  author,  and,  after  having  compared 
it,  article  by  article,  with  the  various  other  publications  that  have  come  to  my 
hands,  I  hesitate  not  to  say,  that  I  think  it  contains  a  greater  amount  of  matter, 
and  a  better  arrangement  of  subjects,  than  any  other  book  I  have  seen.  Your 
rules  and  explanations  are  clear  and  definite,  and  your  examples  are  well  calcu 
lated  to  fix  them  in  the  mind.  I  congratulate  the  community  on  this  valuable 
accession  to  our  list  of  school  books  ;  and  shall  take  pleasure  in  seeing  your 
Arithmetic  extensively  introduced  into  all  our  schools,  as  also  into  that  under  my 
own  care.  "Yours,  with  just  respect,  DATID  P.  PAGX. 

From  the  late  Principal  of  the  Young  Ladies1  High  School,  Boston. 
Dear  Sir  :  I  have  examined  with  great  care  Mr.  Greenleaf  's  National  Arith 
metic,  and  have  used  it  as  a  text-book  for  my  pupils.  In  my  view,  the  plan 
and  execution  of  the  work  are  quite  perfect,  the  rules  being  deduced  analyti 
cally  from  examples,  and  followed  by  copious  questions  for  practice.  The  pupil 
can  hardly  fail  to  understand  as  he  advances  ;  nor  can  he  go  through  the  book, 
without  being  a  master  of  the  science  of  Arithmetic.  This  is  not  an  old  book 
with  a  new  name,  but  the  work  of  one  who  thoroughly  understands  the  subject, 
and  who  has  learned,  from  a  long  and  successful  experience  in  teaching,  how  to 
prepare  one  of  the  very  best  school  books  which  has  ever  been  issued  from  the 
American  press.  Very  respectfully,  E.  BAILEY. 

Having  for  two  or  three  years  past,  made  constant  use  of  Greenleaf  's  Na 
tional  Arithmetic  in  my  School,  I  am  prepared  to  say,  that  it  is  far  superior  to 
any  work  I  have  ever  used. 

It  appears  to  me  to  be  a  complete  system,  and  well  calculated,  not  only  to  in- 
terest  the  pupil,  but  also  to  give  him  a  thorough  knowledge  of  the  science.  I 
think  it  richly  deserves  the  high  commendation  and  liberal  patronage  which  it 
generally  receives.  ALFRED  M.  HOTT, 

Jnst.  Male  School,  Portsmouth,  N.  H. 

I  have  had  the  National  Arithmetic,  by  Benjamin  Greenleaf,  in  use  in  my 
Seminary  for  several  months  past,  and  take  pleasure  in  recommending  it  as  an 
excellent  work. 

I  have  no  hesitation  in  saying,  that  I  not  only  think  it  the  best  single  volume 
on  the  science  of  arithmetic  extant,  but  that  I  consider  its  value  to  be  equal,  if 
not  superior,  to  that  of  any  series  of  arithmetics  now  before  the  American  public. 

D.  RING, 
Principal  of  the  East  Baltimore  Female  Institute. 

From  J.  Peckham,  Esq.,  Teacher,  Westminster,  N.  H. 

B.  Greenleaf,  Esq.  Sir  :  I  take  great  pleasure  in  recommending  your  Na 
tional  Arithmetic.  A  number  of  classes  went  through  with  the  book  in  the 
course  of  my  teaching,  and  I  feel  satisfied  that  they  obtained  a  more  thorough 
and  practical  knowledge  of  the  science,  than  they  would  have  done  by  any  other 
text-book  wiih  which  I  am  acquainted.  While  the  work  is  sufficiently  com 
pendious  and  cheap  for  general  use,  it  at  the  same  time,  fully  illustrates  every 
principle  in  common  business.  I  think  the  appendix  on  book-keeping  a  very 
valuable  addition  to  the  Arithmetic.  Your  obedient  servant, 

JOSEPH  PECKHAM. 


n  reference  to  the  "  Abstract  of  the  Massachusetts  School  Returns,"  for 
1840,  it  will  be  perceived,  that  Greenleaf's  National  Arithmetic  is  used  in 
many  of  the  best  Schools  and  Academies  in  the  State.  And  wherever  teachers 
have  given  this  system  a  fair  trial,  the  result  has  been  highly  satisfactory. 

16 


Robert  S.  Dams1  Publications. 


ALGER'S  MURRAY'S  BOOKS. 


ALGER'S  MURRAY'S  GRAMMAR;  being  an  abridgment 
of  Murray's  English  Grammar,  with  an  Appendix,  containing 
exercises  m  Orthography,  in  Parsing,  in  Syntax,  and  in  Punctua 
tion  ;  designed  for  the  younger  classes  of  learners.  By  Lindley 
Murray.  To  which  Questions  are  added,  Punctuation,  and  the 
notes  under  Rules  in  Syntax  copiously  supplied  from  the  author's 
large  Grammar,  being  his  own  abridgment  entire.  Revised,  pre 
pared,  and  adapted  to  the  use  of  the  "  English  Exercises,"  by  Israel 
Alger,  Jr.,  A.  M.,  formerly  a  teacher  in  Hawkins  Street  School, 
Boston.  Improved  stereotype  edition. 

As  a  cheap  and  compendious  elementary  work  for  general  use,  this  is  pro 
bably  the  best  Grammar  extant,  which  is  indicated  by  its  introduction  into 
many  Schools  and  Academies,  in  various  sections  of  the  United  States. 
Though  furnished  at  a  moderate  price,  it  is  so  copious,  as,  in  most  cases,  to 
supersede  the  necessity  of  a  larger  work. 

fjT  By  a  vote  of  the  School  Committee,  this  work  was  introduced  into  all 
the  Public  Schools  of  the  city  of  Boston. 

ALGER'S  MURRAY'S  ENGLISH  EXERCISES :  consisting 
of  Exercises  in  Parsing,  instances  of  false  Orthography,  violations 
of  the  rules  in  Syntax,  defects  in  Punctuation,  and  violation  of  the 
rules  respecting  perspicuous  and  accurate  writing,  with  which  the 
corresponding  rules,  notes,  and  observations,  in  Murray's  Grammar 
are  incorporated ;  also,  References  in  Promiscuous  Exercises  to  the 
Rules  by  which  the  errors  are  to  be  corrected.  Revised,  prepared 
and  particularly  adapted  to  the  use  of  Schools,  by  Israel  Alger,  Jr., 
A.  M.  Improved  stereotype  edition. 

Extract  from  the  Preface. 

It  is  believed  that  both  teachers  and  pupils  have  labored  under  numerous 
and  serious  inconveniences,  in  relation  to  certain  parts  of  these  Exercises,  for 
the  want  of  those  facilities  which  this  volume  is  designed  to  supply.  Those 
rules  in  Mr.  Murray's  Grammar  which  relate  to  the  correction  of  each  part 
of  the  Exercises  in  Orthography,  Syntax,  Punctuation  and  Rhetorical  con 
struction,  have  been  introduced  into  this  manual  immediately  preceding  the 
Exercises  to  which  they  relate.  The  pupil  being  thus  furnished  with  the 
principles  by  which  he  is  to  be  governed  in  his  corrections,  may  pursue  his 
task  with  profit  and  pleasure.  In  this  edition,  more  than  forty  18mo.  pages 
of  matter  have  been  added  from  Mr.  Murray's  Grammar. 

ALGER'S  PRONOUNCING  INTRODUCTION  TO  MUR 
RAY'S  ENGLISH  READER,  in  which  accents  are  placed  on  the 
principal  words,  to  give  Walker's  pronunciation.  Handsomely 
printed,  from  stereotype  plates. 

ALGER'S  PRONOUNCING  ENGLISH  READER:  being 
Murray's  Reader,  accented  by  Israel  Alger,  Jr.  Printed  from 
handsome  stereotype  plates,  on  good  paper,  and  neatly  bound. 

3pr  These  editions  of  Murray's  books  are  in  the  highest  repute  of  any  othci 
published  in  the  United  States,  and  are  sold  at  a  cheap  price. 

B*  17 


Robert  S.  Davis'  Pullications. 


PARKER'S  EXERCISES  is  ENGLISH  COMPOSITION. 

PROGRESSIVE  EXERCISES  IN  ENGLISH  COMPOSI 
TION.  By  R.  G.  Parker,  A.  M.,  Principal  of  the  Franklin  Gram- 
mar  School,  Boston.  Thirty-ninth  Stereotype  Edition. 

QU*  The  reputation  of  this  little  Manual  is  now  so  well  established  as  to 
render  it  unnecessary  to  present  many  of  the  numerous  testimonials  in  its 
favor,  received  from  teachers  and  others  of  the  first  respectability. 

The  School  Committee  of  Boston  authorized  its  introduction  into  the  Public 
Schools  of  the  city,  soon  after  the  first  edition  was  issued,  and  it  is  now  the 
only  work  on  Composition  used  in  them.  It  has  also  been  adopted  aa  a  text 
book  in  a  large  number  of  the  best  schools  and  higher  seminaries  in  various 
sections  of  the  United  States,  having  been  highly  commended  by  all  intelli 
gent  teachers,  who  have  used  it,  and  the  demand  is  constantly  increasing. 

To  show  the  high  estimate  of  the  work  in  England,  the  fact  may  be  stated,  that 
it  has  been  republishred  and  stereotyped  in  London,  and  nine  large  editions  have 
been  sold  there;  which,  together  with  its  favorable  reception  throughout  the 
United  States,  furnishes  sufficient  evidence  of  its  practical  utility. 

Among  the  public  notices  of  the  work  in  England,  are  the  two  following : 

The  design  of  this  work  is  unejceptionably  good.  By  a  series  of  progres 
eive  exercises  the  scholar  is  conducted  from  the  formation  of  easy  sentences  to 
the  more  difficult  and  complex  arrangement  of  words  and  ideas  He  is,  step 
by  step,  initiated  into  the  rhetorical  propriety  of  the  language,  and  furnished 
•with  directions  and  models  for  analyzing,  classifying,  and  writing  down  his 
thoughts  in  a  distinct  and  comprehensive  manner.  —  London  Jour,  of  .Education. 

Of  the  Exercises  in  Composition,  by  Parker,  we  can  speak  with  unmingled 
praise.  It  is  not  enough  to  say,  that  they  are  the  best  that  we  have,  for  we 
have  none  worth  mention.  The  book  is  fully  effective  both  in  suggesting  ideas 
or  pointing  out  the  method  of  thinking,  and  also  in  teaching  the  mode  of  ex 
pressing  ideas  with  propriety  and  elegance.  —  English  Monthly  Magazine. 

From  Mr.  Walker,  Principal  of  the  Eliot  School,  Boston. 

This  work  is  evidently  the  production  of  a  thorough  and  practical  teacher, 
and  in  my  opinion  it  does  the  author  much  credit.  By  such  a  work  all  the 
difficulties  and  discouragements  which  the  pupil  has  to  encounter,  in  his  first 
attempts  to  write,  are  in  a  great  measure  removed,  and  he  is  led  on,  progres 
sively,  in  a  methodical  and  philosophical  manner,  till  he  can  express  his  ideas 
on  any  subject  which  circumstances  or  occasion  may  require,  not  only  with 
sufficient  distinctness  and  accuracy,  but  even  with  elegance  and  propriety. 
An  elementary  treatise  on  composition,  like  the  one  before  me,  is  certainly 
much  wanted  at  the  present  day.  I  think  this  work  will  have  an  extensive 
circulation,  and  I  hope  the  time  is  not  distant,  when  this  branch  of  education, 
hitherto  much  neglected,  will  receive  that  attention  which  in  some  degree  ita 
importance  demands. 

From  J.  W.  Bulhley,  Esq.,  Principal  of  an  Academy,  Albany. 

I  have  examined  "  Parker's  Exercises  in  Composition,"  and  am  delighted 
with  the  work ;  I  have  often  felt  the  want  of  just  that  kind  of  aid,  that  is  here 
afforded  :  the  use  of  this  book  will  diminish  the  labor  of  the  teacher,  and  great 
ly  facilitate  the  progress  of  the  pupil  in  a  study  that  has  hitherto  been  attended 
with  many  trials  to  the  teacher,  and  perplexities  to  the  laarner. 

If  Mr.  Parker  has  not  strewed  the  path  of  the  student  with  flowers,  he  has 
"removed  many  stumbling-blocks  out  of  the  way,  made  crooked  things  straight, 
and  rough  places  smooth."  It  is  certainly  one  of  the  happiest  efforts  that  I 
have  ever  seen  in  this  department  of  letters,  —  affording  to  the  student  a  beau 
tiful  introduction  to  the  most  important  principles  and  rules  of  rhetoric  ;  and  I 
would  add,  that  if  carefully  studied,  it  will  afford  a  "  sure  guide  "  to  written  com 
position.  I  shall  use  my  influence  to  secure  its  introduction  to  all  our  schools- 

18 


Robert  S.  Davis'  Publications. 


BOSTON     SCHOOL     ATLAS. 


BOSTON  SCHOOL  ATLAS.  Embracing  a  Compendium  of 
Geography.  Containing  seventeen  Maps  and  Charts.  Embellish 
ed  with  instructive  Engravings.  Twelfth  edition,  handsomely 
printed,  from  new  plates.  One  volume,  quarto. 

The  Maps  are  all  beautifully  engraved  and  painted ;  and  that  of  Massachu 
setts,  Connecticut,  and  Rhode  Island,  contains  the  boundaries  of  every  town  in 
those  states. 

O"  Although  this  book  was  designed  for  the  younger  classes  in  schools,  for 
which  it  is  admirably  calculated,  yet  its  maps  are  so  complete,  its  questions 
so  full,  and  its  summary  of  the  science  so  happily  executed,  that,  in  the  opinion 
of  many,  it  contains  all  that  is  necessary  for  the  pupil  in  our  common  schools. 

From  the  Preface  to  the  Sixth  Edition. 

The  universal  approbation  and  extensive  patronage  bestowed  upon  the 
former  editions  of  the  Boston  School  Atlas,  has  induced  the  publishers  to  pre 
sent  this  edition  with  numerous  improvements.  The  maps  of  the  World, 
North  America,  United  States,  Europe,  England,  and  Asia,  hive  been  more 

i»erfectly  drawn,  and  re-engraved  on  steel ;  and  the  maps  of  Maine,  of  New 
lampshire  and  Vermont,  and  of  the  Western  States,  also,  on  steel,  have  been 
added  ;  and  some  improvements  have  been  made  in  the  elemental  part. 

It  has  been  an  object,  in  the  revision  of  this  edition,  to  keep  tne  work,  as 
much  as  possible,  free  from  subjects  liable  to  changes,  and  to  make  it  &  perma 
nent  Geography,  which  may  hereafter  continue  to  be  used  in  classes  without 
the  inconvenience  of  essential  variations  in  different  editions. 

From  R.  G.  Parker,  author  of  "  Progressive  Exercises  in  English  Composi 
tion^  and  other  popular  works. 

I  have  examined  a  copy  of  the  Boston  School  Atlas,  and  have  no  hesitation 
in  recommending  it  as  the  best  introduction  to  the  study  of  Geography  that  I 
have  seen.  The  compiler  has  displayed  much  judgment  in  what  he  has 
omitted^  as  well  as  what  he  has  selected ;  and  has  thereby  presented  to  the 
public  a  neat  manual  of  the  elements  of  the  science,  unencumbered  with  use 
less  matter  and  uninteresting  detail.  The  mechanical  execution  of  the  work 
is  neat  and  creditable,  and  I  doubt  not  that  its  merits  will  shortly  introduce  it 
to  general  use.  Respectfully  yours, 

R.  G.  PARKER. 

From  E.  Bailey,  Principal  of  the  Young  Ladies'  School,  Boston. 
I  was  so  well  pleased  with  the  plan  and  execution  of  the  Boston  School 
Alias,  that  I  introduced  it  into  my  school,  soon  after  the  first  edition  was  pub 
lished.  I  regard  it  as  the  best  work  for  beginners  in  the  study  of  Geography 
which  has  yet  fallen  under  my  observation ;  as  such  I  would  recommend  it  to 
the  notice  of  parents  and  teachers. 

From  the  Principal  of  one  of  the  High  Schools  in  Portland. 
I  have  examined  the  Boston  School  Atlas,  Elements  of  Geography,  &c.,  and 
think  it  admirably  adapted  to  beginners  in  the  study  of  the  several  subjects 
treated  on.     It  is  what  is  wanted  in  all  books  for  learners— simple,  philosophi 
cal,  and  practical.     1  hope  it  will  be  used  extensively. 

Yours,  respectfully,  JAS.  FURBISH. 

I  have  perused  your  Boston  School  Atlas  with  much  satisfaction.  It  seems 
to  me  to  be  what  has  been  needed  as  an  introduction  to  the  study  of  Geogra 
phy,  and  admirably  adapted  to  that  purpose. 

Very  respectfully,  yours,  &c.        B.  D.  EMEBION. 
22 


Robert  S.  Davis'  Publications. 


SMITH'S  CLASS  BOOK  OF  ANATOMY. 


THE  CLASS  BOOK  OF  ANATOMY,  explanatory  of  the  first 
principles  of  Human  Organization,  as  the  basis  of  Physical  Educa 
tion  ;  with  numerous  Illustrations,  a  full  Glossary,  or  explanation 
of  technical  terms,  and  practical  Questions  at  the  bottom  of  the 
page.  By  J.  V.  C.  Smith,  M.  D.,  formerly  Professor  of  General 
Anatomy  and  Physiology  in  the  Berkshire  Medical  Institution. 
Sixth,  Improved  Stereotype  Edition. 

|j*  This  work  has  received  the  highest  testimonials  of  approbation  from 
the  most  respectable  sources,  and  has  already  been  adopted  as  a  text  book  in 
many  schools  and  colleges  in  various  sections  of  the  United  States. 

The  estimation  in  which  it  is  held  in  other  countries  may  be  inferred  from 
the  fact,  that  a  translation  of  it  has  recently  been  made  into  the  Italian  lan 
guage,  at  Palermo,  under  the  supervision  of  the  celebrated  Dr.  Placido  Portel. 
It  is  also  in  the  progress  of  translation  into  the  Hawaiian  language,  by  the 
American  missionaries  at  the  Sandwich  Islands,  to  be  used  in  the  higher 
schools,  among  the  natives ;  and  the  plates  are  soon  to  be  forwarded,  with 
reference  to  that  object,  by  the  American  Board  of  Commissioners  for  Foreign 
Missions ;  which  furnishes  conclusive  evidence  of  its  value  and  utility. 

From  Rev.  Hubbard  Winslow,  Pastor  of  Bowdoin  St.  Church,  Boston. 

Boston,  Nov.  7,  1836. 

I  have  examined  the  Class  Book  of  Anatomy,  by  Dr.  Smith,  with  very  great 
satisfaction.  For  comprehensiveness,  precision,  and  philosophical  arrange 
ment,  it  is  surpassed  by  no  book  of  the  kind  which  I  have  ever  seen.  The 
study  of  Anatomy  and  Physiology,  to  some  extent,  is  exceedingly  interesting 
and  useful  as  a  branch  of  common  education  ;  and  it  is  to  be  desired  that  it 
should  be  more  extensively  adopted  in  all  our  higher  schools.  To  secure  this 
end,  there  is  no  other  book  before  the  public  so  well  prepared  as  the  one  under 
remark.  It  is  also  a  convenient  compend  to  lie  upon  the  table  of  the  scientific 
anatomist  and  physician,  and  a  very  valuable  family  book  for  reference,  and 
for  explanation  of  terms  which  often  occur  in  reading.  TT  ™ 

We  are  gratified  to  see  the  attempt  to  introduce  a  new  subject  to  ordinary 
students.  It  is  wonderful  that  civilized  man  has  been  so  long  willing  to 
remain  ignorant  of  the  residence  of  his  mind,  and  the  instruments  by  which  it 
operates.  The  book  before  us  abounds  in  information  in  which  every  adult 
reader  will  feel  a  deep  interest,  and  from  which  all  may  derive  valuable  les 
sons  of  a  practical  kind.  We  are  gratified  to  see  frequent  references  to  the 
Great  First  Cause  of  life  and  motion.  We  cordially  wish  success  to  his  enter 
prise  in  a  path  almost  untrodden. — American  Annals  of  Education. 

Copy  of  a  Communication  from  Mr.  C.  H.  Allen,  of  the  Franklin  Academy, 
Andover,  Mass. 

North  Andoter,  Dec.  10,  1836. 

Mr.  R.  S.  Davis.  Dear  Sir :  During  my  vacation,  I  have  had  time  to  ex 
amine  Smith's  Class  Book  of  Anatomy,  the  second  edition  of  which  you  have 
recently  published.  I  do  not  hesitate  to  speak  of  it  as  the  very  work  which 
the  public  have  long  demanded.  It  contains  knowledge  which  should  be 
widely  diffused.  The  author  is  remarkably  clear  in  his  explanations  and  des 
criptions,  and  very  systematic  in  his  arrangement.  So  that  he  has  rendered 
this  neglected  branch  of  useful  knowledge  highly  interesting  to  all  classes. 

Yours,  respectfully.  ^  „     , 

J>  CHAS.  H.  ALLIN. 

20 


Robert  S.  Davis"*  Publications. 


FISK'S    GREEK    GRAMMAR,    AND    EXERCISES. 


A  GRAMMAR  OF  THE  GREEK  LANGUAGE,  by  BENJAMIN 

FRANKLIN  FISK.     Twenty-sixth  stereotype  edition. 

The  requisites  in  a  Manual  of  Grammar,  are  simplicity  and  lucidness  of 
arrangement,  condensation  of  thought,  and  accuracy  of  principle  and  expres 
sion.  These  requisites  Mr.  Fisk  appears  to  have  attained  in  a  considerable 
degree  in  his  Greek  Grammar,  of  which  we  have  expressed  approbation  by 
introducing  it  into  our  School. 

FORREST  AND  WYCKOFF,  Principals  of  Collegiate  School,  New  York  City. 

New  York,  October  3d,  1843. 

T  have  used  for  several  years  Fisk's  Greek  Grammar,  and  I  regard  it  among 
the  best,  and  perhaps  the  best,  now  used  in  our  Schools.  Pupils  instructed  in 
it,  encounter  less  difficulty  than  in  others.  E.  H.  JENNY,  A.  M., 

New  York,  October,  1843.  Principal  of"  New  York  Institute." 

Mr.  R.  S.  Davis,  —  I  have  examined  Fisk's  Greek  Grammar,  published  by 
yourself.  To  all  who  will  take  the  trouble  to  examine  it,  its  own  intrinsic 
merit  will  be  its  best  recommendation.  The  Syntax  1  regard  as  decidedly 
superior.  The  rules  are  peculiarly  clear  and  comprehensive,  thereby  relieving 
the  student  from  a  heavy  tax  upon  his  time  and  memory,  to  which  he  would 
otherwise  be  subjected,  and  from  which,  perhaps,  he  is  not  equally  free  in  the 
use  of  any  other  work  of  the  kind. 

C.  TRACY,  Principal  of  Select  English  and  Classical  School. 

New  York  City,  October  28</i,  1843. 
From  Benjamin  Greenleaf,  Esq.,  author  of  the  National  Arithmetic,  etc. 

Bradford,  (Mass.,)  Teacher's  Seminary,  October  16th,  1843.  —  For  several 
years  past,  I  have  used  Fisk's  Greek  Grammar  in  my  seminary.  I  consider  it 
a  work  of  superior  merit.  It  is  well  arranged  ;  and  the  rules  are  clear  and  per 
spicuous.  It  is,  in  my  opinion,  better  adapted  to  initiate  pupils  into  the  idiom 
of  the  Greek  language,  than  any  other  treatise  of  the  kind,  that  I  have  ex 
amined.  FJSK'S  GREEK  EXERCISES  should  be  used  in  connexion  with  the 
Grammar.  A  work  of  this  kind  has  long  been  needed.  It  is  a  production  of 
great  merit.  Yours  respectfully,  BENJAMIN  GREENLEAF. 

Recommendations  fully  concurring  with  the  above  opinions,  have  beei.  received 
from  the  following  gentlemen,  among  many  others,  who  have  recently  adopted 
this  Grammar  in  preference  to  any  other. 
ISAAC  F.  BRAGG,  Principal  of  Male  High  School,  New  York. 


JAMES  N.  McELLioorx, 
WM.  A.  TAYLOR, 
MOORE  AND  FISH, 
CHARLES  W.  FEEKS, 
WASHINGTON  KING, 
J.  JAY  GREENOUGH, 


Mechanics'  Society  School, 
All  Saints  Parochial  School, 
the  New  England  School, 
Classical  and  English  School, 

n  if 

Select  School, 


O3  Fisk's  Greek  Grammar  is  used  in  Harvard  University,  and  in  mani, 
other  Collegiate  and  Academic  Institutions,  in  various  parts  of  the  United  States. 

FISK'S  GREEK  EXERCISES.  Greek  Exercises;  containing 
the  substance  of  the  Greek  Syntax,  illustrated  by  Passages  from 
the  best  Greek  Authors,  to  be  written  out  from  the  words  given  in 
their  simplest  form  ;  by  BENAMIN  FRANKLIN  FISK.  "  Consuetude 
et  exercitatio  facilitatem  maxime  parit."  —  Quintil.  Adapted  to 
the  Author's  "  Greek  Grammar."  Sixteenth  stereotype  edition. 
Fisk's  Greek  Exercises  are  well  adapted  to  illustrate  the  rules  of  the  Gram 
mar,  and  constitute  a  very  useful  accompaniment  thereto. 

(Signed)        J.  B.  KIDDER,  Teacher  of  Select  School,  New  York. 


Robert  S.  Davis"*  Publications. 


LEVERETT'S  CAESAR  AND  FOLSOM'S  CICERO. 


LEVERETT'S  CAESAR'S  COMMENTARIES.  Caii  Julii  Cae- 
saris  Commentarii  de  Bello  Gallico  ad  Codices  Parisinos  recensiti, 
a  N.  L.  Achaintre  et  N.  E.  Lemaire.  Accesserunt  Notulse  An- 
glicae,  atque  Index  Historicus  et  Geographicus.  Curavit  F.  P. 
LEVERETT.  Editio  stereotypa. 

from  John  J.  Owen,  Principal  of  Cornelius  Institute,  New  York,  and  Editor 

ofJCenophon's  Anabasis. 

I  have  examined  with  some  attention  Caesar's  Commentaries,  edited  by 
Leverett,  and  Cicero's  Orations,  edited  by  Folsom,  and  am  happy  to  recom 
mend  them  to  classical  teachers,  as  being,  in  my  estimation,  far  superior  to 
any  other  editions  of  those  works,  to  which  students  in  this  country  have 
general  access.  The  typography  is  fair  and  accurate,  and  the  general  appear 
ance  of  the  books  does  honor  to  the  enterprising  publisher.  I  hope  these 
editions  will  be  extensively  used  in  our  Academies  and  High  Schools. 

(Signed)    JOHN  J.  OWEN.  Cornelius  Institute. 
New  York,  Nov.  22,  1843. 

I  have  attentively  perused  Leverett's  Caesar.  The  neatness  and  accuracy 
of  the  Text,  and  the  beautiful  adaptation  of  the  Notes,  compel  me  to  use  it  in 
preference  to  any  other  that  I  have  seen. 

(Signed)     E.  H.  JENWY,  Principal  of  New  York  Institute. 

New  York,  Nov.  I,  1843. 


FOLSOM'S  CICERO'S  ORATIONS.  M.  T.  Ciceronis  Orationes 
Quaedam  Selectae,  Notis  illustratae.  [By  CHARLES  FOLSOM,  A.  M.] 
In  Usum  Academis  Exoniensis.  Editio  stereotypa,  Tabulis  Ana- 
lyticis  instructa. 

From  Charles  E.  West,  Principal  of  Rutgers  Female  Institute,  New  York. 

I  take  pleasure  in  commending  to  teachers  the  recent  beautiful  edition  of 
Folsom's  Cicero.  The  attractiveness  of  its  text,  notes,  synoptical  and  ana 
lytical  tables,  and  typographical  execution,  led  me  to  place  it  in  the  hands  of 
a  class  of  young  ladies,  who  are  reading  it  with  delight. 

(Signed)     CHARLES  E.  WEST,  Principal  of  R.  F.  I. 

Neio  York,  Nov.  1,  1843. 

I  have  examined  Cicero's  Orations,  edited  by  Charles  Folsom,  and  prefer 
it  to  any  other  I  have  seen.    The   Synopsis  and  Analysis  of  each  Oration  are 
BO  beautifully  given,  that  it  seems  as  a  Rhetoric,  as  well  as  a  Text  Book  for 
learning  Latin.     I  shall  use  it  exclusively  in  the  institution  under  my  charge. 
(Signed)    E.  H.  JENNY.  Principal  of  New  York  Institute. 

New  York,  Nov.  1,  1843. 

I  have  carefully  examined  the  recent  editions  of  Leverett's  Caesar,  and 
Folsom's  Cicero,  and  fully  concur  in  the  opinions  above  expressed. 


(Signed)     WM.  A.  TAYLOR,  Principal  of  All  Saints  Parochial  School. 
ew  York,  Nov.  1843. 

These  editions  of  Ccesar  and  Cicero  are  highly  recommended  by  the  following 
Teachers,  who  have  recently  adopted  them,  in  preference  to  all  others. 
ISAAC  F.  BRAGG,  Principal  of  Male  High  School,  New  York. 

C.  TRACY,  "        "    English  and  Classical  School,        <• 

B.  F.  PARSONS,  <f         "    Female  Classical  School,  " 

W.  MARSH,  "        "  Classical  and  English  School,  Lyceum,  Brooklyn. 


Robert  S.  Davis1  Publications. 


WALKER'S  SCHOOL  DICTIONARY  AND  THE  CLASSICAL  READER. 

WALKER'S  BOSTON  SCHOOL  DICTIONARY.  Walker's 
Critical  Pronouncing  Dictionary,  and  Expositor  of  the  English  Lan 
guage.  Abridged  for  the  use  of  Schools  throughout  the  United 
States.  To  which  is  annexed,  an  Abridgment  of  WALKER'S  KEY 
to  the  pronunciation  of  Greek,  Latin  and  Scripture  Proper  Names. 
Boston  stereotype  edition. 

jtjT  This  handsome  and  correct  edition,  prepared  for  the  Boston  schools, 
with  great  care,  has  so  long  been  used,  that  it  is  only  necessary  for  the  pub 
lisher  to  keep  it  in  a  respectable  dress,  to  ensure  it  a  general  circulation. 

The  price  of  the  work,  neatly  bound  in  leather,  is  reduced  to  50  cts.  single, 
$5,00  a  dozen. 

THE  CLASSICAL  READER.  A  Selection  of  Lessons  in 
Prose  and  Verse,  from  the  most  esteemed  English  and  American 
Writers.  Intended  for  the  use  of  the  higher  classes  in  Public  and 
Private  Seminaries.  By  Rev.  F.  W.  P.  Greenwood  and  G.  B. 
Emerson,  of  Boston.  Tenth  stereotype  edition. 

This  work  is  highly  approved,  as  a  First  doss  Reader,  and  has  received 
many  commendable  notices  from  Public  Journals  throughout  the  United 
States,  Irom  which  the  following  are  selected. 

From  the  Visiter  and  Telegraph,  Richmond,  Va. 

This  work  is  a  valuable  acquisition  to  our  schools.  It  is  a  work  purely 
national  and  modern.  It  has  many  valuable  historical  facts  and  anecdotes  in 
relation  to  the  early  history,  the  character,  manners,  geography  and  scenery 
of  our  country.  In  the  matter  it  contains,  it  is  well  adapted  to  the  taste,  feel 
ings,  and  habits  of  the  present  age.  It  embodies  many  of  the  brightest  and 
most  sparkling  gems  of  Irving,  Webster,  Everett,  Jefferson,  Channing,  Sparks, 
Bryant,  Percival,  &c. 

From  the  American  Journal  of  Education, 

We  are  happy  to  see  another  valuable  addition  to  the  list  of  reading  books, 
—one  which  has  been  compiled  with  a  strict  regard  to  the  tendency  of  the 
pieces  it  contains,  and  which  bears  the  stamp  of  so  high  a  standard  01  literary 
taste.  In  these  respects  the  Classical  Reader  is  highly  creditable  to  its 
editors. 

Extract  from  ike  North  American  Review. 

The  Classical  Reader  is  selected  from  the  very  best  authors,  and  the  quan 
tity  from  each,  or  the  number  of  pieces  of  a  similar  character,  by  different 
authors,  affords  all  that  can  be  required  for  classes,  and  in  sufficient  variety, 
too,  of  manner,  to  facilitate  greatly  the  formation  of  correct  habits  of  reading, 
and  a  good  taste.  From  each  of  those  considerations,  we  give  it  our  cordial 
recommendation. 


fjT  The  Publisher  respectfulfa  solicits  the  attention  of  Teacher's,  School 
Committees,  and  all  interested  in  me  cause  of  Education,  to  the  foregoing  list 
of  School  Books, — -feeling  confident  that  an  examination  of  the  works  will  lead 
to  a  conviction  of  their  merits, — copies  of  which  will  be  furnished  for  this  pur 
pose,  with  a  view  to  their  adoption,  without  charge. 


YB  35288 


GREENLEAF'S    ARITHMETIC. 

TENTH  IMPROVED  STEREOTYPE  EDITION. 


THL  NATIONAL  ARITHMETIC,  on  the  Inductive  Systen 
combining  the  Analytic  and  Synthetic  Methods. 

$3*  '* ';*-.  •uirjiuion  of  Teachers,  a  ,  t,ed  in  thorough  education,  in  invi 

to  this  work.  Having  already  been  adopted  in  many  of  the  best  Seminaries  in  vari 
sections  of  the  United  Si  ales,  it  has  been  highly  recommended  by  all  teacheia  v 
have  used  it.  The  following  OPINIONS  of  the  work,  are  selected  from  nearly 
hundred  communications,  addressed  to  the  Aytlior  and  Publisher. 

Fro ,t  E.  Bailey,  Esq.,  Aut-hor  of  «'  First  Lessons  in  Algebra,"  to  the  publisher. 

^eat  care  Mr.  Greenleal's  National  Arithmet] 

and  use  it  as  a  text-book  for  my  pupils.  In  rny  view,  the  plan  and  execution  of 
work  are  quite  perfect,  the  rules  being  deduced  analytically  from  examples,  and 

3nd\ 

he  auvarues  ;   nor  can  he  go  through  the  book,  without  b<£ng  a*  master  of  the  sci 
of  Arithmetic.     This  is  not  an  old  book  with  a  new  name,  but  the  work  of  one 
thoroughly  understands  the  subject,  and  who  has  learned,  from  a  long  and  success! 
experience  in  teaching,  how  to  prepare  one  of  the  very  best  school-books  which  H 
ever  been  issued  from  the  American  press.  Very  respectfully, 

Mineral  Spring  School,  Lynn,  May  18?^,  1839.  "     E.  BAILEY.! 


Benj 

in  trod  n 


£sq.     Dear  Sir  :     Having  examined,  and,  to  , 
•uls  the  National  Arithmetic,  of  which  you  are  the  auti 
.ue  public,  no  less  than  to  yourself,  to  express  our 
The  method,  arrangement,  and  quantum  i»f 

its  rules  are  demonstrated,  togel 
with  ii.-  belief,! 

the  patronage  of  every  lover  oi 

(Signed,)        llA/.kv  PICKERING.  V  -, 

A.  M.   ilOYT, 

JAMES  HOYT,  EDWARD  J.  LAIGII 

Portsmouth,  Aug.  5,  1838.  POTTER,  j 

te  of  a  iMerfrom  Rev.  Dr.  Hopkins,  President  of  William's  College. 

My  opinion  of  Greenieaf  's  Arithmetic  is,  that  it  is  adapted  to  give  a  more  tl 

:>(  that  science,  than  any  otiier  that  I  have  seen.     Respectfully,  y< 

Dec.  SO,  1837.  M.  HOPKINS. 

•';l  Institute,  ; 
V   National 

s  u| 

'(ing  the  host  of  so 
•'ive  it  would  l)e  equally  [< 
.  I  shall  cheerfully  introduce  it  into  n 

:".  M.  i  j.  r. 


contair 
less. 

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Pkil 

I   have   examined,   with 

Aritlmi'eiic  by  B.  Green  1«  ,u.     J  can  sav,  witlioi. 
plete  and  well-arranged  School  System,  in  tin? 
calculated  than  any  other  to  pif-j 
suits  where  a  kn<>wl< 
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cfs  will  he  found  in  the  Advertising  Sheet  at  th«  close  of '  t'r. 
and  sold  by  the  principal  Books  filers  throng  h'mt  the  United  States. 


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